Prose is, simply put, any writing, or the language used in writing it, that is not poetry. Prose adheres to the rules and conventions of spelling, grammar, punctuation, and usage, and it is characterized by how it follows natural speech patterns. The simplest complete unit of prose is the sentence, and we communicate using prose every day.
In sharp contrast, poetry is a carefully constructed form of expression, often† characterized by how it veers away from natural speech patterns. The simplest complete unit of a poem is the line, and poetry does not always adhere to the rules and conventions that prose must. Although we likely encounter forms of poetry every day, it is not a natural form of communication; after all, when was the last time you held an entire conversation in verse or waxed poetic with everything you said?
The purpose of prose is clear communication; the purpose of poetry is quite varied, depending on who you ask‡. In general, though, poetry tends to be more concerned with exploring the abstract (such as emotions) without necessarily explaining it°, while prose tends to be more focused on conveying information as quickly and effectively as possible.
Now, there are forms of writing that blur the line between poetry and prose, and one of them in particular, the prose poem, is likely the source of the prose/poem term confusion.
Prose poems look, at first glance, to be prose; unlike most poetry, prose poems are structured into sentences and sometimes paragraphs, much like regular writing, and often follow the grammatical conventions of prose. That's where the similarities between prose and prose poetry end, however. Although a prose poem lacks the hallmark line structure of more typical poems, the actual content of a prose poem remains unmistakably poetic, be it through the use of consistently strange syntaxes and imageries or other poetic devices like rhythm, alliteration, consonance, dissonance, internal rhyme, etc.
On the opposite end of the continuum, you have prose pieces, such as sketches and vignettes, that contain strong poetic elements (such as uniquely rich imageries) and/or lack typical prose elements (such as a "point" to the piece), but more importantly lack the heavy use of poetic devices and artificially constructed structures found in poetry, which is why they are classified as prose.
- Prose and poetry are quite different, and one should never refer to a poem or the language used to construct it as "prose".
Prose is essentially all forms of writing that are not poetry, though it also refers to the language used to construct such pieces. It is derived from natural speech patterns and adheres to the rules and conventions of the language.
The simplest complete unit of prose is the sentence.
Poetry is a carefully constructed, and therefore artificial, form of writing. It often† does not adhere to natural speech patterns or language conventions.
The simplest complete unit of poetry is the line.
There are some forms of writing which blur the distinction between prose and poetry. One of these is prose poetry.
Prose poetry, although it may look like prose on the surface, is a type of poetry, hence the "poetry" part of the name.
* Or "prosaic", if you prefer.
† But not always, like almost everything about poetry.
‡ And what their mood is at the time.
° Now watch me get mobbed by poetry people for writing such an inaccurate/grossly overgeneralized statement.