Fill a small cup or plastic bag with some of the cat's favorite treats. Being rewarded with treats teaches the cat to enjoy having its claws trimmed.
Hold the cat around its midsection with one hand. With the other, take one of its paws, and squeeze one of the toe pads lightly until the claw extends. Note the pink wedge that extends part way down the claw. This is the quick of the claw. Avoid cutting into the quick or cutting the claw off too close to it.
Place the clippers over the claw, about midway between the start of the quick and the sharp edge of the claw. Cut off the tip of the claw to blunt the edge. Cutting any lower than the tip can potentially cause a cat great distress. Repeat the process on the rest of the cat's claws. After the nails have been trimmed, reward the cat with a treat to associate the trimming with a positive experience.