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!