When you start to learn to play mandolin [1], people usually teach you to play open chords, ones where some of the strings are left open to ring out, primarily because they're easier. This gives you a full vibrant sound, and I like it for accompanying singing, but I don't think it's a good match for contra dancing. I think playing chords with no open strings works a lot better because by slightly lifting up your fingers you can mute it [2]. This lets you play the chord really short and punchy, and you can alternate this with fully muted chords for a percussive sound:

• boom-chuck.mp3

What if you only know open chords? How do you start doing this? Because we're not using any open strings, you can learn a very small number of chord shapes and then play any chord you want by combination of chord shape and which fret you play at. I use only three chord shapes: barred A, barred Am, and barred E:

    barred A:              barred Am:              barred E/Em:
      E --|--|--|--M--|-     --|--|--M--|--|-      --|--M--|--|--|-
      A --|--|--|--X--|-     --|--|--X--|--|-      --|--X--|--|--|-
      D --|--X--|--|--|-     --|--X--|--|--|-      --|--X--|--|--|-
      G --|--X--|--|--|-     --|--X--|--|--|-      --|--|--|--X--|-
             *                    *                     *

(The 'X's are fingers, the 'M' a muted string. The '*' marks the fret is where I think of the chord as being set.)

If you learn that on the 7th fret you have D, Dm, and A, that's really all you need to figure out all the others. Or, if you like tables, I've made one, bolding the ones I use the most:

Fret		Barred A		Barred Am		Barred E
2		A		Am		E or Em
3		Bb		Bbm		F or Fm
4		B		Bm		F# or F#m
5		C		Cm		G or Gm
7		D		Dm		A or Am
9		E		Em		B or Bm

There are other chord shapes, but you can go a long way with just these three. I'm only just starting to use others in my playing.

There are various rhythms you can make out of alternating chords and muting. Some examples:

• boom-chuck.mp3 (band -- nightingale)
• four-one.mp3 (band -- free raisins)
• one-three-four.mp3 (band -- free raisins)
• half-time.mp3 (band -- great bear trio)
• half-time+four-one.mp3
• one-two.mp3

I especially like this as a complement to piano.

Update 2014-08-09: A followup.

[1] Most of this holds for guitar too, but I've been playing mostly mandolin lately.

[2] Muting is also possible with the right hand, but I know nothing about this.