Fingerstyle guitar basically means that you're plucking the strings with your fingers, instead of you using a pick. I do have a pick; I'm using a thumb pick here as opposed to a flat pick. That just suits my style fine. You can use a thumb pick or sometimes people like to use just their bare thumb, flesh on the strings.
Fingerstyle is probably the oldest way of playing stringed instruments, including guitar. It stems from, probably, the classical genre on up through today, including people like Chet Atkins and Merle Travis that use a thumb pick like I do, and then on into jazz, like people like Lenny Breau.