Popular poetry types include haiku, free verse, sonnets, and acrostic poems. It's one thing to define each type; it's another to enjoy a sample platter.
The structure of a poem refers to the way it is presented to the reader. This could include technical things such as the line length and stanza format. Or it could include the flow of the words used and ideas conveyed. Line length shows the reader how it should be read.
Literature that is not poetry is focused more on guiding you where it wants you to go through every day, conversational logic. Poetry uses figurative language. It plays on the senses and emotions of its reader, which means that it doesn't play by the same rules and its words carry more weight.