The parts of a poem are, to an extent, dependent upon what form it takes. Many modern poets write using free verse, which does not use consistent patterns of rhythm or form.
However more traditional forms of poetry have more distinct form and structure:
A stanza, or verse, is a grouped set of lines within a poem, usually separated from other stanzas by a blank line.
There are different poetic forms based on sets of visual rules which determine how they are made up, for example:
Sonnets- a fourteen line poem following a set rhyme scheme and logical structure. Made infamous by Shakespeare, and particularly associated with love poetry. They often end in what is known as a "rhyming couplet"
Here's a cool example of a sonnet read out loud:
Haiku- a short form of Japanese poetry traditionally made up of 3 lines of 5, 7 and 5 syllables or "onji".