To cut your cat's nails, hold one of his paws in your hand.  One by one, hold each of his toes and lightly push into his toe pad in the direction of his main paw pad.  This will cause his claw to rise out of his paw.  Gently cut the end of the nail, which will be sharp.  If you look closely at the semi-transparent nail you will see a pink base that stretches out into the main nail.  This is called the quick and you do not want to cut into this as this will cause the nail to bleed.  If for some reason you do accidentally cut into it, you should have a styptic pencil available, which you pack into the nail to control the bleeding.

A wonderful site that explains this thoroughly with illustrations is How to Trim Your Cat's Claws.