What music did The Beatles play?

they played a sort of alternative rock depending on the instrument

If you were looking for a general "catch-all" classification, probably pop or rock.

They covered a lot of musical ground including Rock and Roll, Rock, Hard Rock, Country, Blues, Ballads, Psychodelia, Instrumentals, Soul, Folk, Indian, ie. not stuck in the rut of just one musical genre.

The Beatles' music influenced their contemporaries, and those who came after them. It appealed to all age groups and to people all over the world.