Achieving this technique takes a lot of practice and familiarity with the fingerboard. The most common thing to tap is an arpeggio, which is a chord broken up into its individual notes.
Anchor the first finger of your left hand on the lowest of the notes you wish to tap and hold it there. The middle note will be taken care of by one of your other left hand fingers, usually the fourth finger, and the highest note is covered by your right hand. Different guitarists use different fingers, and some even use the edge of the pick.
Eddie Van Halen uses different patterns, but a good one to start with is a descending pattern from the highest note. Place both fingers of your right hand in their places before you tap firmly with your right hand on the highest note. Release that note, then release the middle note.
Tap the high note again, placing both your left hand fingers at the same time, and repeat the pull offs again.
A double stop is basically two notes played at the same time, generally in parallel motion. Think about two parallel runs set apart from each other starting and ending at the same time, but on different scale degrees.
Eddie Van Halen does this quite a lot, whether he sets the notes in thirds, fourths or sixths. Again, fingerboard familiarity is necessary. Starting at the high end of a major scale, descend on one string--for example the third string, playing frets 12,11,9,7,5,4,2 and open. At the same time, play the first string descending the same scale starting on the 12th fret, and play 12, 10, 8, 7, 5, 3, 2 and open.
When you play them at the same time, you end up with a descending G major scale played in parallel sixth double stops.
These are played the same way as double stops, except Van Halen sort of "arpeggiates" them by playing one note at a time instead of two. It comes across as wide leaps, but the effect is the same. Play the above exercise one note at a time, alternating between strings, and you'll get the idea. He'll often slide in between notes on either or both strings to add a little zest to the line, and you can as well.
To escape the boredom brought on by block chords, Eddie Van Halen makes frequent use of arpeggiated chords in the main riff sections of songs. It's pretty easy to do, although you'd have to be Eddie to get it just like him. Once you know the chord progression, play up and down the strings, palm muting as you see fit to accent or dull a note here and there. Play with different arpeggio patterns to find what suits your own style.
As a side note to the arpeggio section, Eddie also connected the arpeggiated chords with a small scale run down or up to the next bass note. He uses this technique especially heavily in the song "Cathedral." Work this in rhythmically, so as not to interrupt the flow of the sound.