Rhapsodies were originally a literary form. Epic poems, but also collections got that name from the 16th century onwards. At the end of the eighteenth, but mostly in the nineteenth century, it was also used as the name of a piece of music consisting of freeform variations. The rhapsody was mostly developed by Franz Liszt, and this Bartók rhapsody is very much inspired by his work.
For Bartók this was the first among many compositions using folk tunes. Contrary to what the opus number might suggest, this was not his first work. The publisher might have called it opus 1, but this was already the third ‘list’ of works by the composer. He wrote three versions of it: one for piano, one with orchestra and one with two pianos.