Having a will allows you to decide the people that will take from your estate after your death as opposed to leaving it up to state law. Another reason to have a will is that it lets you pick your personal representative and allows you to avoid the cost of a personal representative's bond required by probate court. If you pass without a will the court will appoint the PR for your estate. A third reason to have a will is that if you have minor children, you can name their guardian in your will.
A living will is part of an advance directive for health care. The living will portion provides that if you have a terminal condition, become persistently unconscious or have an "end-stage" condition you may instruct that your life is not to be extended by life-sustaining medical procedures.
An alternative to a will is the revocable trust. Having a revocable trusts allows for your distributions from your estate to occur outside of the probate court system. Further, a revocable trust is not filed with the court. As it is a private document you are not listing all of your property in a public record.