Just save the bottom (the heart) of your lettuce. Place lettuce heart in a small container of water with the cut side facing up. Set in a sunny spot, and change the water every few days. New leaves and roots should form within a few days. The old, cut leaves may turn brown and withered, and should get removed.
You can keep it in the container, or transplant it outside. Harvest by picking off the outer leaves, leaving the inner leaves untouched. As long as the heart remains undamaged, it should produce new lettuce.
You can also do this with celery and green onions. I always keep the stubs of green onions, and regrow them inside and outside. Really, no one should live without having a container of perpetually growing lettuce or green onions. It's that easy.