Re: free verse vs prose
"Free verse" occupies that part of the prose/poetry continuum which has "blank verse" on the one side and "prose poetry" on the other.
Blank verse is an unrhymed sequence of iambic pentameters (e.g. as in Milton's Paradise Lost). Other similar forms are not very common in English poetry; but examples in German might be the unrhymed tetrameters of Goethe and Heine. (I exclude unrhymed Greek and Latin verse, as this is constructed by length of syllable, not stress.)
Prose poetry on the other hand has the heightened diction, repetitions, rhythmic recurrences, emotive content, etc. that we are accustomed to find in poetry, but it is presented as prose and organised by the paragraph, not by the line or block of lines. An example would be the Dream Fugue of De Quincey; cf. also the prose poems of Baudelaire and Rimbaud.
Between them you find "free verse". This is sometimes fully or partly rhymed, and loosely based upon standard metrical units (as in the free verse of T.S. Eliot); but more commonly the rhythm varies from line to line, or at least between groups of lines, as in the Cantos of Ezra Pound, or the poems of William Carlos Williams and D.H. Lawrence.
The line-breaks are significant in free verse, by the way, because once you discard regularities of rhythm and rhyme, you are left with very few ways of emphasising particular words or phrases. Words at the beginning and end of a line seem more significant to the reader; and when free verse is read aloud, line-ends are also where the natural pauses occur (another form of emphasis).
The result is that writers of free verse tend towards a presentation in which the distribution of lines reflects the syntax.
Alternatively, they may well insert line-breaks in unusual places, for effect.
Not a professional ESL teacher.