A vegetarian diet is considered to be "healthier" by a subjective standard than meat-eating diets, but this assumption suffers from low scientific testing. One recent 11.6 year-study found that ordinary people who followed a vegetarian diet were less likely to suffer or die from heart disease; this study only considered the exact figures, not correlation or causation. However, it seems likely that vegetarian diets are overall healthier; just the switch from meat to plants will lower consumption of fats, cholesterol, and sodium, all of which are indicated in many health issues.
The most important aspect of a vegetarian diet (not a vegan diet, which excludes all animal products such as milk and eggs) is making sure nutritional requirements are met. Most strict vegetarians who follow nutritional guidelines have little trouble with nutritional deficiencies, while others who are more lax find that they are missing important nutrients such as amino acids and certain vitamins. All of this is more to do with knowing what is in the food, not simply by avoiding meat.
Probably the best aspect of a vegetarian diet is the low intake of saturated (and increasingly trans) fats, which are indicated for heart disease as well as other health issues. While plant products contain fats, they usually don't contain them in the high quantities of meat.
Finally, one very important aspect is that consuming a largely plant-based diet reduced the amount of overall calories consumed. Plant products are high in fiber and a larger amount contains fewer calories than an equivalent amount of meat. This means that vegetarians are less likely to suffer from obesity, as long as their diets are managed according to nutritional guidelines; it is very easy to be a vegetarian and make bad food choices. Obesity aggravates many other health problems, and a vegetarian diet can decrease the risk of obesity-related diseases.
The website wisegeek.com has a solid overview of scientific studies done over the years, but the studies themselves should be examined for accuracy.