Secondary dominants are a great device for giving your chord progression more 'forward movement', incorporating non-diatonic notes into (mostly) diatonic chord progressions, and creating interesting harmonies in your songs and arrangements.

In essence, each diatonic chord in a chord family can be 'set up' with a dominant chord a perfect fifth above it's root. So a 'D' chord (whether it's minor or major) can be 'set up' by playing an A7 chord before it (A is a perfect 5th above D). This is demonstrated in the key of C Major in the diagram below. The top line are the secondary dominant chords, and the bottom line are they chords they resolve to.

You'll notice that the roots of the secondary dominant chords are all still diatonic to the key of C, and in fact most of the chords themselves remain diatonic - most of the time only one note in each secondary dominant chord is non-diatonic. We can therefore turn any diatonic chord into a dominant chord, and it will still function in the diatonic key. The one exception to this is really the IV7 chord, (F7) which resolves outside the key (to the bVII chord).

I've written these out on a treble stave. Notice that there are two options for resolving to the seventh degree - from the IV7 to the bVII, or from the #IV to the VII. However, both of these are fairly problematic; as we observed earlier the IV7 resolves outside of the tonality of C major, and the half-diminished VII chord doesn't really provide a satisfying resolution from the #IV. (Also, the #IV7 can easily sound out-of-place)

We can take this a step further by voicing each of these chords over their thirds, which creates a satisfying assending chromatic movement in the bass...

EXAMPLES OF SONGS THAT USE SECONDARY DOMINANTS

TAKING IT TO THE NEXT LEVEL!

When we use a secondary dominant, we are temporarily modulating into another key. We can increase this effect by 'setting up' the secondary dominant with other chords from that key. These are usually chords that have a 'sub-dominant function' in the new key - typically chords ii or IV. We see this a lot in the chord changes of many jazz standards. 

In 'Alice in Wonderland' we are in the key of A. The secondary dominant III7 chord (C#7) is used to modulate into the key of F#m, and it is set up by the ii7 chord of this key (G#m7b5).