You can finger pick a guitar either with or without finger picks. Some players just use their fingers and the tip of their fingernail to get a smoother sound. In addition to the medium you use to finger pick, you also have a couple of different techniques:

Finger picking in general requires an up-down movement: down with your thumb and up with your fingers. The thumb leads the bass rhythm, while the other fingers play the melody. Depending on how many fingers you use, you can also play different styles.

The Travis style, which Dale McCoy demonstrates on Folkways and many other players of folk and bluegrass tend to use, requires picking with mainly the thumb and the index finger, while resting the pinky on the pick guard while you are playing. Some players will use a slight variation of this and add the middle finger.

Players of traditional classical guitar typically use four fingers: thumb, index finger, middle finger and ring finger, with no support on the guitar.