Left-handed drinking is the worst drinking game I've been involved in.
Basically everyone (assuming they're right handed) has to drink only with the left hand. If you forget, i.e. caught drinking right-handed you have to finish off your drink there and then.
Sounds easy, but the fun is everyone will forget at sometime + you can set-up mates etc. to catch them out. (Best one is getting the round in and saying cheers; someone always forgets which hand they're drinking with and they have to down a full pint it in one.)
Variants include alternating drinking hands each round. Basically, you get ****ed real quick, so not for the faint-hearted.