II-V-I progressions are commonly used in jazz music, but also can be found in many pop and rock songs. If the progression appears in a jazz context, the chords are often played as seventh chords with extensions and alterations.
The II, V and I chords of a key are very good at helping our ears identify the key centre of a piece of music. You will come across this chord progression in many songs from Abba through to Queen.
This chord progression combines the previous two progressions. It starts out with a I-VI chord progression, then moves to the II-V-I progression outlined above. Basically the II chord is replacing the IV chord.
The dominant chord (V chord) creates such a strong resolution to the I chord that we can approach any chord in a progression with it's dominant 5th. These dominants are known as secondary dominants since they do not occur in the original key, but can still be used in a progression to add more variation.
For example, if we take the following I-VI-II-V-I progression:
We can approach each of these chords with it's dominant chord, as follows:
As you can see, we have added in a variety of dominant 7th chords that are not in key, but the progression still works because of the strong relationship between the dominant chord and the chord that follows it. The example above then resolves using the original dominant of the key, E7, to A Major.
Secondary dominants may in fact be played as triads, dominant sevenths, or dominant sevenths with extensions (although eleventh chords are rare). An example of a song which uses this trick is 'Don't Know Why' by Norah Jones.
These chord progressions are of course not all that exist, but are by far the most common. Any chord diatonic to the key will sound good with any other (see Harmonizing the Major Scale). Experiment with different chords and see if you can come up with some progressions of your own.