Alliteration, assonance and consonance are all figures of speech specifically referring to arrangements of words by sound. Since what distinguishes poetry from prose is regular sonic structures added on top of grammatical structuring of language, all three of these figures are used regularly in poetry. In free verse, especially, absent regular rhyme and meter, these figures are often the core of sonic structuring.
Consonance is a figure in which the same consonant sound is used multiple times in a line. Alliteration is a type of consonance, restricted so that the repetition of the consonant occurs in the first letters of successive words (this was the basis of Anglo-Saxon “alliterative” strong stress poetry). Assonance is the repetition of vowel sounds.
To analyze the use of these devices. Look at two significant words (nouns or verbs rather than prepositions or articles) that both, for example, start with the same letter. Assume that the poet is using the alliteration to indicate that there is some special connection between the two words, and think about what the significance of the connection might be.