This is the most commonly asked question I get when people find out I am vegetarian. I am lacto-vegetarian which means I eat all plants and dairy products but no animal flesh. There are a number of other kinds of vegetarians such as ovo-lacto- (eat eggs and dairy), and vegans who eat no animal products whatsoever.
The idea that a vegetarian would not get enough protein is a big myth. Today I found an article about the protein myth that nicely summarizes my feelings about this – how it is true that vegetarians actually do get enough protein and are far healthier than non-vegetarians.
Anyway, I recommend reading the whole article as well as the links to Dr. Dean Ornish which are quite interesting.