Poetry is carefully selected words that are designed to evoke an emotion. Poetry is different from prose, or normal novel and short story writing, because it has a specific rhythm created by how the words are put together into lines and stanzas.
The first thing you should do when analyzing a poem is read it to yourself. You can read it in your head and then out loud. This will help you get familiar with the language. You can also take notice of the form. Is the poem a sonnet, a ballad, or a haiku? Count the number of stanzas and lines. You can also try looking the poem up online to see what form of poem it is.
More important than the type of poem you have is the poem’s rhythm. Read the poem out loud several times, paying attention to rhyme and other sound devices such as alliteration (repeated initial sounds like “Peter Piper picked a peck of pickled peppers”) and onomatopoeia (words that look like they sound, such as “crash” and “bang”). These are important because they help create the sound of the poem, and in poetry sound is just as important as meaning because it helps make the meaning.
Consider this example, from "The Raven" by Edgar Allen Poe.
Once upon a midnight dreary, while I pondered, weak and weary,
You can tell that dreary and weary are rhymed, which creates a slow and moody poem.
Next you can move to more sophisticated analysis such as speaker, tone, and mood. The speaker is the character from whose perspective the poem is told. The mood is the emotional atmosphere of the poem, which is created by the tone (word choice that shows author’s attitude toward the subject). Is the poem happy or sad? Is it energetic or slow?