Today many new guns are Over Bored as well as back bored. Any gun shop with a gunsmith on site, should be able to measure it as there is a standard Choke Gauge, again as Tim pointed out, which might not be exactly that accurate regarding what performance you might get. If it is an old gun the dime trick should be (use a new shinny dime!
What kind is it? Most fixed choke shotgun have a choke designation at the beginning of the barrel somewhere. "$" for skeet etc.
Performance wise, things are getting technical on the new shotguns. Back Bore, Over Bore, Lengthened Forcing Cones. If the shotgun is not a classic and has enough metal in the barrel, you can get it threaded for screw in chokes pretty reasonably. Then you not only will know what you have, but can change the constriction to what ever you want.
If you go to any of the new screw in choke suppliers , (Briley, Muller, Carlson, Trulock etc.) they will list what the various restrictions are. Today there are a lot more chokes available then in the past. Spreadeer, Cylinder, Light Skeet, Skeet, out to Full and Extra Full. I have screw in chokes now in all my shotguns except my early Browning Citori 4 Barrel Skeet set, which all 4 barrels are SK/SK.