Prose and poetry are two ways of classifying types of writing. Sometimes we group literature according to the topic matter (fiction, non-fiction, mystery, drama, and so on), but here we're grouping it according to format. Fortunately, the distinction between what is prose and what is poetry is usually pretty instinctive.
Let's start with the term that's probably more familiar: poetry. Poems are often easy to spot because the author is usually working to create a unique structure involving all of the senses. For instance, the poet may arrange words very deliberately on the page to look a certain way; poets choose words not only for their meaning, but also because they rhyme or sound out a special rhythm; some poets even limit themselves to a fixed number of words or syllables. Even punctuation and grammar can be manipulated by the poet to fit in with the poem's structure.
In contrast, the word prose comes from a Latin word meaning straightforward, and that's what prose writing is like. The story being told might be complex, but you can count on all the features of everyday speech, such as paragraphs, sentences, and all the usual punctuation. Types of literature that usually fall into the prose category include most essays, short stories, novels, and plays.
Just to keep you on your toes, be aware that authors sometimes dabble in a combination of both forms. For example, Shakespeare is known for switching back and forth within his plays, having certain characters speak in poetry and others in prose.