Verse has multiple meanings in the context of poetry. It's a synonym for "stanza," a unit comprising a set of lines—or rarely, one line—set apart from other units by line spaces. More generally, it used to differentiate a poetic work from one in prose, e.g., verse drama á la Shakespeare. Broader still it can refer to poetry in general as opposed to prose or other forms of writing.
According to Merriam-Webster, "verse" is derived from "Middle English vers, fers, in part borrowed from Anglo-French vers, verse in part going back to Old English fers, both borrowed from Latin versus 'furrow, measure of land, row, line, line of writing, line of metrical writing,' action noun derived from vertere 'to cause to turn, rotate.'"
Note that "turn" and "turning" are at the root of this word. It is also fascinating to realize that plowing and thus farming lie hidden under "verse" and therefore under poetry itself. Poetry then is related to cultivation and fertility, explaining in part why so many religious rituals and much scripture itself are expressed in the forms and language of poetry.
Versus, as noted above, shares its etymology with "verse" in the sense of a turn or the act of turning. In common usage it means against or opposed to, a denotation not inapplicable to poetry. Though sadly many people turn away from poetry, finding it obscure or too difficult, many others turn to it for consolation, for inspiration, for a new perspective on the world, for experiencing the power of charged language. In the best sense, poetry stands opposed to the ordinary, the mundane, the quotidian, "news that stays news" as Ezra Pound aptly states it.
A term from the world of books and publishing, verso shares its etymology with the two terms discussed above. A verso is the left-hand page of a book or journal, facing a right-hand or recto page. Merriam-Webster relates its specific etymology: "New Latin verso (folio) the page being turned." While relatively few books of poetry qualify as "page-turners," the fact that most poetry is consumed in book or journal form—though the availability of so much poetry, great, good, mediocre, bad, awful, on the Internet may well be giving print a run for its money. One hopes that online reading of poetry supplements that of print consumption. Truly the best reading is tactile and even olfactory and there's nothing quite like holding a physical book and slowly reading a poem, running one's eyes over shapely verses and lingering over especially compelling words.
One might say that a book of poems, with its recto and verso pages, is the ultimate flowering from its roots in vertere, the plow's turning the earth, the tractor coming to the end of a row and turning to make a new furrow, working against the soil to turn it into new growth, moving from left to right and back again, all to bring new growth that readers will turn to again and again.