Based on the definition of vegetarian as a diet which excludes consumption of animal flesh, technically, yes, eggs are vegetarian, since they are not animal flesh (the meat, muscles or tissue of an animal). Many vegetarians include eggs in their diet, while still abstaining from eating chicken, cows, pigs, fish and all animals. According to terminology, you're an "ovo-vegetarian", that is, a vegetarian who eats eggs.
Note that while eggs are vegetarian, they are certainly, under no circumstances, considered vegan.
See also: Is milk vegetarian?
However, in some parts of the world, particularly in India, eggs are indeed considered to be meat, and some vegetarians will not eat them. Usually, this is tied to a religious practice, such as Hinduism. Vegetarian Hindus, and those who have come to vegetarianism from a Hindu background consider eggs to be meat and do not include them in their vegetarian diet. So, while most Westerners believe that eggs are vegetarian, many Easterners do not; this means that whether or not eggs are vegetarian is really dependent on a cultural agreement.
Eggs Come from Chickens -- How Can They Be Vegetarian?
Vegetarians avoid eating dead animals, and while eggs are not dead animals, there's some debate as to whether or not vegetarians should avoid eating foods that require killing animals, even if they are still avoiding the animal flesh itself. For example, caviar eggs are obtained from slicing open fish stomachs and taking the eggs from inside the fish.
The eggs themselves are vegetarian, but the animal must die in order to obtain them.
See also: Is caviar vegetarian?
Do chickens have to die to lay eggs? Well, no, not really. But the truth is, they do. Unless you're buying eggs from your next door neighbor and have seen the conditions of their farm, you are supporting industrialized factory farming, which kills millions of chickens while obtaining their eggs each year.
If this is a concern for you, you'll probably want to be fully vegan and avoid milk and dairy products as well as eggs. But not to worry. There's plenty of ways to substitute eggs in cooking and baking, and it can be as simple as buying a commercial egg replacer. And of course, there's plenty of vegan milk replacements and dairy substitutes you can use as well. There's even dyeable vegan Easter egg substitutes these days!
See also: Easy vegan recipes to try
What About Health Concerns?
If you're eating more meatless meals to reduce your fat intake or lower your cholesterol, eating eggs might not be your best bet. The American Heart Association recommends that adults should eat no more than four eggs per week, while the Australian Heart Foundation is ok with up to six per week, but if you're trying reduce your fat or cholesterol intake, it's best to eat fewer.
Want to try cooking without eggs? For starters, try out an egg-free vegan quiche or one of these popular easy tofu scramble recipes to substitute for eggs for breakfast. Or, if you want to avoid tofu, try out this new VeganEgg which you really have to see (and taste!) to believe. For Easter, check out these vegan Easter egg substitute ideas (including one vegan Easter "egg" you can even dye just like regular eggs).
So Should I Eat Eggs if I'm a Vegetarian?
The bottom line: The important thing is not whether or not a food item fits within your (or another person's) definition of vegetarian or not, but whether or not you feel that consuming a food is morally justifiable or not.
If you're trying to be kind to animals, you may want to explore going vegan. If you're eating vegetarian for your health, you'll likely find that you want to reduce the number of eggs you eat. Try out an egg replacer and a few vegan recipes, and see what you think!