Your appendix is a small, closed tube that is about the size of your finger. It attaches to the beginning of your large intestine, where the small and large intestines meet. It is open at the end that connects to the large intestine and closed at the other end, so material can move into and out of the appendix but it has nowhere to go. The appendix can contract like the small intestine, and it also has a secreting lining like the small intestine.
If the open end of the appendix gets plugged for some reason -- either because of swelling or because something from the large intestine gets stuck in the opening -- then the appendix starts to swell because of the secretions from the lining. The swelling shuts off the blood supply and the appendix tissue dies. An operation to remove the swollen appendix is the way this problem gets fixed. If it isn't removed, the appendix eventually bursts, which can be fatal if untreated.
It does not appear that the appendix does anything. People traveling to foreign countries or on expeditions will sometimes have it removed to prevent the possibility of a problem.
Here are some interesting links: