On the
guitar, finger tapping is the act of plucking a string with your right
hand middle finger (the most common "tapping finger"), either open or fretted, and then hammering on the fretboard with the same finger. Usually you will do
hammer-ons/
pull-offs with your left
hand in between the time that your right hand is tapping.
Tapping can be very simple,
e.g. 2 finger tapping where you are simply playing a fast trill (like Toccata and Feuge in D Minor), or it can be very complicated like Buckethead's 10 finger tapping.
Many people claim that Eddie Van Halen invented multiple finger tapping, others say it was Billy Sheehan or Randy Rhoads, etc. But it was definitely Van Halen who brought it to the main stream audience in the 80's.