I assume throughout that you can read tablature.
While Eddie Van Halen is generally credited as being the player who popularized this technique, some form of it has existed at least since Paganini. The technique is actually pretty widely used now, and Emmett Chapman created an instrument known as The Stick specifically to employ tapping techniques. The Stick is usually an 8 to 12-string instrument. It superficially resembles a guitar, but it's basically a big vertically oriented fretboard intended to be played with various two-hand tapping techniques.
I'm going to provide a few simple exercises and two easy tapping riffs to get you started with this legato techique. This is all for guitar, as it's the only stringed instrument that I know how to play. Most of it should apply to to bass guitar as well.
First, get out your guitar. I suggest practicing on an acoustic guitar. Tapping is usually an electric guitar kind of thing, but if you can perform a techique well on an acoustic it will usually be effortless on electric. That said, use whatever guitar you like. You might not play or even own an acoustic, and it's certainly not necessary to learn techniques acoustically. Get comfortable; hold your instrument however you normally would. Try to practice the following in this position. It'll be a lot easier to use these techniques within a song if you don't have to rearrange yourself to employ them. If you're already proficient at any section, feel free to skip ahead. If you can already tap, then this is pretty worthless to you.
1.1: The Hammer On
The hammer on is a ubiquitous technique in Rock music, and it will serve you well, particularly if you're into playing solo material. A hammer on played by simply picking one note, and then fretting a new note without picking. Here's an example:
e------
B------
G-4h5--
D------
A------
E------
To play this, fret the G string at the fourth fret with your index finger. Pluck the string to sound the note. Now quickly use your middle finger to fret the string at the fifth fret, but without releasing your index finger. The action of fretting with your middle finger is performed as a strike, which gives the technique its name. There you go; a very simple hammer on.
You should practice this with all four fingers:
e--------5h8---
B-----5h7---5h9
G--5h6---------
D--------------
A--------------
E--------------
That 5h9 might be a bit of a reach. Speed and long reaches are the point of tapping. It's good for you. If you absolutely can't do it, don't worry. It may be impossible for some people on some instruments. Work on it though; It will come up again later.
1.2: Chromatic Hammer On
Chromatic, in this case, just means playing down a given string, moving one fret at a time. Another way to do this is to slide, but this is about tapping.
A chromatic hammer on is performed in much the same manner as a regular hammer on, but with all four fingers in a row. Here's your example:
e---------
B--5h6h7h8
G---------
D---------
A---------
E---------
Fret the B string at the fifth fret. Pluck the string, and then perform the technique from 1.1. This time, however, repeat he maneuver with your ring and then pinky fingers. You should hear four distinct notes sound. Practice this until you can do it.
You can also skip frets:
e---------
B---------
G--5h6h8--
D---------
A---------
E---------
This is like the last example, but you skip the use of your ring finger. Since you're skipping frets now, try this:
e--5h6h8h9
B---------
G---------
D---------
A---------
E---------
These hammer on patterns, and others, can be done anywhere. I picked the fifth fret as a general starting place since it's far enough apart that things aren't too easy, but still close enough that almost anyyone should be able to reach without hurting him or her-self. Practice them all over to get a feel for doing a hammer on all over the neck.
2.1: The Pull Off
The pull off is another common soloing techique that occurs everywhere in guitar based music. Like the derivative and integral, they are inverse operations.
Here is your first(?) pull off:
e---------
B--6p5----
G---------
D---------
A---------
E---------
To perform this pull off, fret the fifth fret of the B string with your index finger. This is the same position as a hammer on now, but don't play anything yet! Fret the string at the sixth fret, with your middle finger. Do not release the index finger; you should be holding the string down in two different frets with two different fingers now. Once you are doing this, pluck the string to sound it, and before it dies release your index finger by pulling it away from the fretboard. You should hear the pitch of the note drop noticeably.
If you are noticing a volume drop, this is normal. To minimize this, practice pulling down on the string with your index finger and then releasing. Don't do this too wildly, or you'll hear the note bend out of pitch. Think of it as plucking the string lightly with your fretting hand. This will take some practice.
2.2: Chromatic Pull Off
This is exactly the same as the 1.2, but in reverse:
e---------
B---------
G--8p7p6p5
D---------
A---------
E---------
Place all four fingers on the fretboard first. Pluck the fretted string, and then pull off each finger in reverse order: pinky, ring, then middle. You can pull off with the idex finger as well to sound an open note at the end if you wish. Make sure you can make the pull off sound with each finger. Use the fret-hand picking technique discussed in 2.1 if necessary.
Again, just like a hammer on, you can skip frets:
e---------
B---------
G--8p6p4--
D---------
A---------
E---------
e--9p8p6p5
B---------
G---------
D---------
A---------
E---------
3.1: Combining the techniques
Once you play one note, you can continue to play some interesting things with nothing but hammering on and pulling off. Try these little examples without cheating (Use only one actual string pluck to get started):
e---------
B---------
G-5h6p5h6-
D---------
A---------
E---------
Just repeat that for a while. See if you can keep it going. Once you can do that, try it with other fingers:
e---------------
B---------------
G--5h6p5h7p5h8p5
D---------------
A---------------
E---------------
This one will be a little harder to keep going without picking. If you've got that down, attempt this chromatic riff:
e---------------
B--5h6h7h8p7p6p5
G---------------
D---------------
A---------------
E---------------
3.2: Finally, some songs (a little bit)
The first example is a Cardigans song called A Good Horse.
e-----------------------
B-----------------------
G------4----------------
D--2\4---2p0-----0h2p0-0
A------------2-2-------2
E----------------------3
Here's another one, this time the Rolling Stones song Paint it, Black. I find it a little bit harder than the last song.
e-----------------------------2------------
B--5-7-8-10-8-7-5-5-4-5-7-5/2---2h3h5p3p2h3
G------------------------------------------
D------------------------------------------
A------------------------------------------
E------------------------------------------
4.1: Tapping
Now for what this write up-is really about. Tapping is nothing more than hammers and pulls on the fretboard with the picking hand. Some people, like Van Halen, do some pick-juggling and use the index finger; other players prefer to maintain their grip on the pick and use the middle finger to do it. Some people are exceptionally good and use four or more fingers on each hand to play tapping riffs. I play fingerstyle, so it's just as easy for me to use my index, but I can't do eight-finger yet.
In order to tap, just strike a string with your chosen tapping finger. Don't use you're entire hand though; that way leads madness and a lack of precision, not to mention tendon problems. Cradle the top and bottom of the neck with your picking hand. Gently rest the thumb on top of the neck, by the low E string. Touch, but do not press or hold, the bottom of the neck, near the low e string, with your pinky (and ring and middle, if desired). You should only be moving the finger you are tapping with, the rest of the hand is just there for support, and to mute the high e, if necessary. Make sure to strike the string hard enough that it sounds, but don't hit it so hard that you damage your finger. You will also have to do a tiny pull off each time when you release to get a note to sound properly, just like when you were doing regular a regular pull off.
To get started tapping, try this: (A 'T' above a note means that you should tap it with your picking hand.)
T
e---------
B--12-5h7-
G---------
D---------
A---------
E---------
If you're having trouble playing that, here's a further explanation: Fret the B string with your left hand, at the fifth fret. Now strike the string with your right hand, at the twelfth fret. This should sound a B, one octave higher than the open string. Now release with your right hand finger, and an E note should sound. While that note is sounding, hammer down on the seventh fret with your ring finger to play an F#. Repeat this and try to build speed.
You can do that a little bit differently:
T
e---------
B-12-7p5--
G---------
D---------
A---------
E---------
This is basically the same as last time, except the notes descend each time. Notice that both of these would be very difficult to do quickly without the tap.
You don't have to tap just for reach; you can also do it purely in the name of speed:
T
e---------
B---------
G--7-5h6--
D---------
A---------
E---------
Repeat that very quickly. You can probably play it much faster by tapping than just using hammer ons and pull offs. Tapping can be combined with any combination of hammer on and pull off techniques, for a lot of interesting possibilities. Experiment!
4.2: A Couple Songs to Practice
The first is part of Bach's Toccata and Fugue in D minor:
Play this first:
T T T T T T T T
B--------------------------------------------
G-14-12-14-10-14-9-14-7-14-6-14-7-14-9-14-10-
T T T T T T T T
B--------------------------------------------
G-14-2-14-4-14-6-14-7-14-6-14-7-14-9-14-10---
Play this second:
T T T T T T T T
B-15-13-15-11-15-10-15-8-15-7-15-8-15-10-15-11-
G----------------------------------------------
T T T T T T T T
B-15-3-15-5-15-7-15-8-15-7-15-8-15-10-15-11----
G----------------------------------------------
I don't actually play that with pull-offs, but there are some places you could use them. I'll leave that to you, as practice. You might notice that somtimes it sounds like you're playing two notes at once. This is because when you tap, the string vibrates on both sides of the fret. This isn't very noticable when your fretting hand is close to your tapping hand, but when they get farther apart you can hear it. It's especially noticable if you tap without fretting something. To reduce this, practice muting with your fretting hand when your tap strikes, and quickly fretting as you release the tap.
The second song here is the one that popularized tapping in the first place: EVH's Eruption.
Repeat each part to taste. I generally play each part four times, and the last one a couple extra. Note that there is actually more to this piece, but I think this is probably the most recognizable part.
T
B-9-2h5
T
B-10-2h5
T
B-10-4h7
T
B-12-4h7
T
B-12-5h9
T
B-13-5h8
T
B-15-7h10
T
B-17-9h12
End on a hammered 17, and hold it for a bit. I like to add a little vibrato on the end. Yes, the whole thing is on the B string.
There are plenty of more complicated examples out there, but these are a couple good ones to get you started. They require good use of the technique, but they aren't terribly difficult either. You may also want to practice sounding hammer ons and pull offs without using a pick to start out. If you can master that, you can play some very piano like pieces that involve simultaneous lead and bass lines. Once you master these, be sure to find other two handed tapping riffs to practice on!
Lots of bands use two hand tapping, and plenty of examples can be found on any tab site by searching for the names given in Chris-O's write-up, among others. One of my personal favorites that's a little hard to find is KMFDM, particularly as performed by Joolz. Let me know if you find out how to play the two hand tapping riff in Hau Ruck!