The mandolin is an accepted and much loved part of folk and country music, adding the characteristic double-course treble to a tune either as single note picking or the sometimes cliched tremolo that reminds one of the origins of the instrument in Naples. (Think of 'O Sole Mio'). Mandolins are also great chord instruments for accompaniment playing straightforward rhythm or chopped chords (strummed and then stopped to give a more staccato effect).
Mandolins come in a variety of styles which generally reflect their use in different types of music. In truth, the styles and the music are interchangeable but their construction and sound quality make them better suited to certain music.
Bluegrass and Country favour the f-style while folk musicians favour the a-style. The old original bowl-back is favoured by classical players.