Poetry is often described as 'literature in metrical form' or 'compositions forming rhythmic lines'. It has a set of specific features that differentiate it from other forms of literature very clearly. It is not necessary that all the elements are always employed in every poem that is penned. However, the presence of at least two of these elements is noted in most poems. For example, a lot of poets choose to write "blank verses" which are poems that lack rhyme. However, a blank verse does have a set rhythm and meter pattern that is followed. Now, a "free verse" will neither have rhyme or rhythm, but these verses are usually opulent in other elements like metaphors, symbols and spectacular word images. So, poetry writing offers a lot of scope for experimentation when it comes to choosing literary elements or devices as per the needs of the poet.
A stanza is to a poem what a paragraph is to a piece of prosaic writing - a fixed number of lines of verse forming a single unit of a poem. A poem is usually composed of multiple stanzas that are separated from each other an empty line in between. Usually, all stanzas are made up of equal number of lines in a single poem. However, there are many examples of poems where this approach has been majorly deviated from. A poem may have a combination stanzas that have varying number of lines.
Based on the number of lines present in a stanza, they are assigned different names. They are:

~ A "couplet" is a stanza that has only 2 lines.

~ A "tercet" is composed of 3 lines.

~ A "quatrain" consists of 4 lines.

~ A "cinquain" has 5 lines.

~ A "sestet" comprises 6 lines.

~ A "sonnet" is an entire poem with exactly 14 lines.
Example of A Couplet
"True wit is nature to advantage dress'd;
What oft was thought, but ne'er so well express'd." - From Alexander Pope's "An Essay on Criticism"
Examples of A Tercet
"furu ike ya
kawazu tobikomu
mizu no oto" - Haiku by Matsuo Bashō

roughly translating to:

"An old silent pond...
A frog jumps into the pond,
splash! Silence again."
A Quatrain

"The curfew tolls the knell of parting day,
The lowing herd wind slowly o'er the lea,
The plowman homeward plods his weary way,
And leaves the world to darkness and to me." - From Thomas Gray's "Elegy Written in a Country Churchyard"
Example of A Cinquain
With faint dry sound,
Like steps of passing ghosts,
The leaves, frost-crisp'd, break from the trees
And fall." - From Crapsey's "November Night"
Example of A Sestat
"It was many and many a year ago,
In a kingdom by the sea,
That a maiden there lived whom you may know
By the name of ANNABEL LEE;
And this maiden she lived with no other thought
Than to love and be loved by me." - From Edgar Allan Poe's "Annabel Lee"
Example of A Sonnet
"When I consider how my light is spent
Ere half my days, in this dark world and wide,
And that one talent which is death to hide,
Lodged with me useless, though my soul more bent
To serve therewith my Maker, and present
My true account, lest he returning chide;
"Doth God exact day-labor, light denied?"
I fondly ask; but Patience to prevent
That murmur, soon replies, "God doth not need
Either man's work or his own gifts; who best
Bear his mild yoke, they serve him best. His state
Is Kingly. Thousands at his bidding speed
And post o'er land and ocean without rest;
They also serve who only stand and wait."" - John Milton's "On His Blindness"
Rhyme and Rhyme Scheme
Rhyming in poetry is one convention that makes this form of literature recognizably different from prose and drama. Even in this age when free verses are quite popular, rhyme in poesy is what renders it poetic. A very unique quality of rhyme in poetry is that it has the ability to provide a systematic flow to a bundle of thoughts that may seem absolutely chaotic if put together otherwise.
A rhyme smoothens out the rough edges and abrupt protrusions. This element can be simplistically defined as the similarity in the sounds of two or more lines. In poetry, this is generally achieved by using similar sounding words at the end of lines.
Example of a Rhyme
"Humpty Dumpty sat on a wall,
Humpty Dumpty had a great fall.
All the king's horses and all the king's men
Couldn't put Humpty together again."
Here, 'wall' and 'fall' sound alike by virtue of the similitude in their vowel sounds and so does 'men' and 'again'. Only the initial consonant sound differs. It would be interesting to note at this point that words like 'wall' and 'fall' that rhyme without an effort are called true rhymes; slant rhymes are those words that do rhyme but with a little effort and some poetic licenses being granted. 'Men' and 'again' is a good instance of slant rhyme usage.
The function of a rhyme extends beyond giving poetry its identity. It helps give structure to all the themes that a poet wishes to cover in a particular peace. When 2 lines, which may or may not be consecutive to each other, rhyme, it mostly indicates a cohesive thematic bond between them.

"Two roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveler, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;"
In these lines from Robert Frost's "The Road Not Taken", the highlights words clearly rhyme with one another. If you skip the second line, and go on to read the first, third and the fourth lines one after the other, they will make complete sense to you. This is simply because the poet deviated from the basic point a tad with the second line right after what he said in the first. The second line is parenthetical in nature, almost like an aside. The word 'both' does not rhyme with either 'wood', 'stood' or 'could', at least not very obviously. Here the rhyme scheme was devised in a manner where the reader is enabled to establish the direct link between certain lines of expression and trace the continuity. The deviation was very intentional. Now, this is often the case but not always. In this very stanza itself, you will see that 'both' rhymes with 'undergrowth', but there isn't a direct link between these two lines.

That brings us to just another purpose of rhyme. When you have the repetition of a sound at least twice in a poem, it serves the purpose of clubbing a certain thought expressed in a single stanza together so that the poet can move on to a different but related line of thought in the next stanza. So, when Frost rhymed the second line with the fifth one in this stanza, his purpose of portraying this verse as a unified whole was served. He would then move on from what he saw in the first stanza to what he did and why in the second - a clear albeit small departure from the first point.
Both the functions stated above stand true for rhyme deviations opted for in the various stanzas of a single poem as well. That is where the rhyme scheme comes in. Take the first two stanzas from Robert Browning's "The Last Ride Together".

"I said--Then, dearest, since 'tis so, (a)
Since now at length my fate I know, (a)
Since nothing all my love avails, (b)
Since all, my life seemed meant for, fails, (b)
Since this was written and needs must be-- (c)
My whole heart rises up to bless (d)
Your name in pride and thankfulness! (d)
Take back the hope you gave--I claim (e)
Only a memory of the same, (e)
--And this beside, if you will not blame, (e)
Your leave for one more last ride with me. (c)

My mistress bent that brow of hers; (f)
Those deep dark eyes where pride demurs (f)
When pity would be softening through, (g)
Fixed me a breathing-while or two (g)
With life or death in the balance: right! (h)
The blood replenished me again; (i)
My last thought was at least not vain: (i)
I and my mistress, side by side (j)
Shall be together, breathe and ride, (j)
So, one day more am I deified. (j)
Who knows but the world may end tonight? (h)"
The rhyme scheme follows the same pattern in both the stanzas, the fifth and the eleventh lines rhyming. But, see the alphabets next to each line - aabbcddeeec and ffgghiijjjh. The frequency and order of the occurrence of alphabets match in both the stanzas, but the alphabets in the two stanzas do not match!

Lastly, there are cases when rhyming words exist in a single line itself. In such a case, it's called middle or internal rhyme. For instance, take these line from "Don't Fence Me In" written by Cole Porter:

"Just turn me loose let me straddle my old saddle,
Underneath the western skies,
On my cayuse let me wander over yonder,
'Til I see the mountains rise."
Once you know the scheme a poet has chosen to use, you'll be able to analyze and comprehend why he has used the scheme he has.
Rhythm and Meter
The primary thing to keep in mind here is that 'rhyme' and 'rhythm' are not the same at all. Rhythm is basically the pattern in which a poet chooses to sequence the stressed and unstressed syllables in every line of a poem, for the creation of oral patterns.
The three factors that help determine the rhythm in a poem are:

~ The total number of syllables present in each line.
~ The total count of accented (stressed) syllables in each line.
~ The tally of recurring patterns of two or three syllables - stressed and unstressed - clubbed in every line.

Each recurring pattern is individually called a foot. And a number of feet, on identification, can tell us the systematic rhythm or the meter that a poem follows.

In poetry, a stressed syllable is tagged with a "/" and an unstressed one is marked with a "U". There are various types of foot and they are named accordingly.

One foot: Monometer
Two feet: Dimeter
Three feet: Trimeter
Four feet: Tetrameter
Five feet: Pentameter
Six feet: Hexameter

And there are five different types of constant beat patterns that the feet can occur in:

Iamb (Iambic) - One weak syllable followed by one accented syllable.
Trochee (Trochaic) - One accented syllable followed by one weak syllable.
Anapæst (Anapæstic) - Two weak syllables followed by one accented syllable.
Dactyl (Dactylic) - One accented syllable followed by two weak syllables.
Spondee (Spondaic) - Two consecutive accented syllables. This can usually be found at the end of a line.
(The upper-cased portions are indicative of the stressed or prominently lifted syllables)
A Iambic Pentameter
"Nor FRIENDS | nor FOES, | to ME | welCOME | you ARE:
Things PAST | redRESS | are NOW | with ME | past CARE." - From William Shakespeare's "Richard II" (Act II, Scene 3)
A Trochaic Tetrameter
"SHOULD you | ASK me, | WHENCE these | STORies?
WHENCE these | LEGends | AND tra | Ditions,
WITH the | ODours | OF the | FORest,
WITH the | DEW and | DAMP of | MEAdows," - From Henry Wadsworth Longfellow's "The Song of Hiawatha"
An Anapæstic Hexameter
"The imMOR | tal deSIRE | of imMOR | tals we SAW | in their FAC | es and SIGHED." - From W. B. Yeats's "The Wanderings of Oisin"
A Dactylic or Heroic Hexameter
"THIS is the | FORest prim- | Eval. The | MURmuring | PINES and the | HEM locks" - From Henry Wadsworth Longfellow's "Evangeline"
A Spondee
"Arma vir | Umque can | O, TroI | aE quI | prImus ab | OrIs"
dactyl |dactyl | spondee|spondee|dactyl | spondee - From Vergil's "Aeneid"
Alliteration is the repetition of a particular consonant or a vowel sound in the initial stressed syllables of a series of words or phrases in close succession.
"I have stood still and stopped the sound of feet."
- From Robert Frost's "Acquainted with the Night

"Fly o'er waste fens and windy fields."
- From Alfred Tennyson's "Sir Galahad"

"Behemoth, biggest born of earth, upheaved His vastness"
- From John Milton's "Paradise Lost: The Seventh Book"

"For the sky and the sea, and the sea and the sky."
- From Samuel Taylor Coleridge's "The Rime of the Ancyent Marinere"

This element is not used in every poem. But, when used, it is basically employed because lines with alliteration roll of the tongue in a manner that accentuates the beauty of the thought expressed. It adds to the rhythm of the poetry in ways very pleasing for the reading.
Simply put, a simile is a direct comparison drawn between two concepts, objects, or people using a verb like 'resembles' or connectives such as 'like', 'as' or 'than'.
"O my Luve's like a red, red rose
That's newly sprung in June;
O my Luve's like the melodie
That's sweetly play'd in tune."
In "My Love is Like a Red, Red Rose" by Robert Burns, the poet uses two similes in the very first stanza itself. First, he compares his love for his beloved to a freshly blossomed red rose to express how perfect it is, untainted in any way. Second, he compares his feelings to a soulful melody that is played to perfection. In both cases, the poet has tried to stress on how the inherent purity and beauty of his love renders it perfect.
A Birthday
Another very good example of a poem with profuse usage of similes is Christina Georgina Rossetti's "A Birthday".

"MY heart is like a singing bird
Whose nest is in a water'd shoot;
My heart is like an apple-tree
Whose boughs are bent with thick-set fruit;
My heart is like a rainbow shell
That paddles in a halcyon sea;
My heart is gladder than all these,
Because my love is come to me."
Metaphor is an indirect parallel drawn between two completely unrelated things. It is a comparison, yes, but metaphors do not use the connectives 'like', 'as' and 'than'. A metaphor usually has more layers and depth than a simile which in the resemblance is usually more linear. Any metaphor can also have multiple interpretations depending on how complicated the poet chooses to make it.
"Hope is the thing with feathers
That perches in the soul,
And sings the tune--without the words,
And never stops at all,"
In Emily Dickinson's poem "Hope", the poetess describes hope in the form of a bird. Like a bird sings at times whether it is happy and sad, similarly, hope springs eternal in the mind of man. Like a dismal bird chooses to vent its grief through a wordless tune, hope soothes always soothes the battered, morbid soul of a grieving man in order to replenish ebbing vitality. One may not know where is hope springs from, just like the unknown words to the tune a bird sings, but its present is always felt by man in times of happiness and sorrow.
The figurative painting of a vivid picture in the mind of a reader with words is imagery. This element is most exploited in descriptive poem where the poet has the scope to use ornate adjectives, lofty language and an exquisitely elaborate canvas to give wings to his imagination. Of course, this scope is primarily offered by the dynamic nature of a descriptive poem.
In Samuel Coleridge's deft description the gardens in Xanadu in his poem "Kubla Khan: or, A Vision in a Dream" is an appropriate instance of imagery usage.

"So twice five miles of fertile ground
With walls and towers were girdled round:
And here were gardens bright with sinuous rills,
Where blossomed many an incense-bearing tree;
And here were forests ancient as the hills,
Enfolding sunny spots of greenery."
Since imagery can be used to appeal to each of the five human senses, there are five different kinds of it that has been used by poets over the ages. They are:

Auditory imagery
Gustatory imagery
Olfactory imagery
Tactile imagery
Visual imagery
The presentation of a tangible object that actually represents an abstract or intangible concept or idea is symbolism. A symbol can be presented to the readers in the form of a character, an object strategically placed in the narrative, a word or phrase, or even a place. A symbol is mostly subtle in nature or at least never blatantly explained. Symbols are mostly multi-layered in nature and can be interpreted differently by different people. Over the years, owing to repeated usage, some objects have acquired one particular value that is usually associated with them, like the apple is usually seen as a symbol of seduction and sensuality (the forbidden fruit association), the loss and regrowth of leaves in a tree has come to be seen as the circle of life, the raven is indicative of imminent death and so on. However, none of these associations must be considered to be absolute or taken for granted by a reader as the presentation of these very objects can change massively depending on the context of poems.

"Ah Sun-flower! weary of time,
Who countest the steps of the Sun:
Seeking after that sweet golden clime
Where the travellers journey is done."

Here, William Blake uses sunflowers to represent human beings who yearn to escape to a higher spiritual plane but are unable to do so for they are shackled by a material existence, just like the flowers which can only look up to the sun expectantly but are deeply rooted to the earth in this lifetime. This, however, is only one of the interpretations of this particular poem. The sunflowers symbolism has been read in different ways by different literary experts.
All these elements of poetry are fluid tools at his disposal of a poet that he can bend and customize in order to convey his ideas most effectively.