Poetry relies very heavily on the use of figurative language to create images and sounds. Many poems also present themes or lessons. In order to appraise a poem critically, it is necessary to analyze the figurative language that is present. For example, does the poet use similes, metaphors, or other "visual" imagery that helps the reader to see or otherwise make connections? Some types of figurative language create sounds that make the poem particularly effective; one example is "The Bells" by Edgar Allan Poe. The vowel sounds in the poem echo the sounds that ringing bells would make. An effective criticism of any poem focuses on the rhyme scheme, if present, and the rhythmic nature of the language if applicable. Finally, the message that the poem carries is significant. An example is "Ozymandias" by Percy Bysshe Shelley; by the end of this brief poem the message that all humanity is nothing more than dust in the wind is transmitted via a powerful image of an ancient city that is no more.
You want to look at several aspects of a poem to do an analysis of it. The first is the poetic techniques that it contains. Does the author use metaphors, imagery, symbolism, paradox? Then, look at the structure. Is it metered and rhymed, or free-style? Another thing to look at is to compare it to other poems written by the same author, or other poems of a similar style (for example, if it is a haiku, compare it to other haiku poems; if a sonnet, compare it to other sonnets). One last thing to look at is the poem's meaning, and how effectively the word choices and structure convey that meaning.
I have provided a link below that goes into more detail on how to analyze a poem. Good luck!