Prose is so-called "ordinary writing" — made up of sentences and paragraphs, without any metrical (or rhyming) structure.
If you write, "I walked about all alone over the hillsides," that's prose. If you say, "I wondered lonely as a cloud/that floats on high o'er vales and hills" that's poetry. See the difference? (Let's not get into prose poetry!) From prose we get the term prosaic, meaning "ordinary" or "commonplace," or lacking the specially delicacy and beauty of its supposed opposite — poetry.