Choke is a constriction towards the muzzle of the barrel. When referring to choke, people almost always are talking about shotguns, even though some rifles too have a slight choke in their barrels to improve accuracy...
Back to shotguns - as you might be aware, shotguns are usually used to fire multiple projectile cartridges, see
http://en.wikipedia.org/wiki/Birdshot for more.
To control how these projectiles "pattern" some distance after exiting the barrel, chokes are used. Choke constriction usually varies from zero (true cylinder or no choke) to full choke... in between you have chokes like Improved Cylinder (IC), Modified Cylinder (MC) etc.
In some cases you might find extra open chokes, in which case the choked end of the barrel is actually a little wider than the rest of the barrel, and also extra tight chokes which are tighter than full chokes - like extra full choke, or super full choke...
As you would imagine, the tighter the choke - the less the spread of shot for a given distance from the barrel and vice versa. Many people incorrectly claim that tighter chokes increase the range of the gun - this is COMPLETELY FALSE. Choke only controls the spread of shot.
Of course if you are shooting at shorter ranges, you would want to use more open chokes as you would like a greater spread of shot at closer range - therefore giving you a much better chance of hitting the target; conversely if you are shooting at longer distances you would want the shot to travel further away before spreading out... Which is probably why some people cling to the myth of tighter chokes = longer range!
Earlier, all shotguns used to come with fixed chokes - that is, the choke was part of the barrel itself and therefore could not be changed based on changing shooting needs. Although one can get a good gunsmith to modify a tighter choke into a less constrictive one - such a change would be PERMANENT and if the gunsmith mucks up the job, the damage would not be easy to rectify, you'd probably need new barrels...

AFAIK it is NOT possible to get a gunsmith to "tighten" an open choke, say from true cylinder to MC...
These days many of the new shotguns models sport "screw-in" chokes on one or both barrels. This greatly increases the flexibly of the gun, as you can change chokes, selecting the most appropriate one for the job at hand.
HTH
Cheers!
Abhijeet