The motivation for people to become vegetarian can range from ethical/moral (believing it is cruel to subject animals to butchering) to nutritional (simply believing that a vegetarian diet is healthier) to environmental (the higher ecological costs of raising livestock versus grains/vegetables). My spouse has been a vegetarian for fifteen years and got started on a dare, bet by a friend that she couldn't eat meat free for a week.
Most individuals, and especially most Americans, consume far too much meat in their diets. The United States government RDA (Recommended Daily Allowance) for meat consumption is six ounces in one day. Three ounces is approximately the size of a pack of playing cards. So the next time you sit down for that half pound burger or eight ounce steak, realize that you are, in that one meal, consuming more meat in one sitting than the US government recommends you consume in an entire day.
Non-vegetarians will often argue that it is impossible to have an adequately balanced diet without meat, but the truth of the matter is that there are many other efficient sources of protein: eggs, beans, nuts, peanut butter, soy, cheese, yogurt, and other dairy products. It is simple to incorporate these into your dietary regimen, but one must make a deliberate and conscientious effort to make sure that protein is included with every meal. It takes a little more thought and planning, but it is possible to have a protein rich diet without meat.
There are also an abundance of fake meat products out on the market, made with everything from tofu, tempeh, and various grains that contain protein. Mornigstar, Boca Burger, and Gardenburger all have extensive lines of fake meat products that have evolved along way from the traditional funky tasting tofu hot dogs of twenty years ago.
Many of these are available at your major supermarket chains, with your local organic food store likely to have a more extensive selection. And before you stereotype these products is inedible, hippie-vegetarian products, try a few of them, you may find that you like them more than their meat counterparts.
Numerous studies have shown that diets containing high amounts of meat (especially red meat) are likely to contribute to higher cholesterol, incidents of atherosclerosis (hardening of the arteries), and other risk factors for congestive heart failure. When you consider that heart disease is by far the number one killer of Americans, and that many incidents of heart deaths could have been prevented with simple lifestyle changes such as proper diet and exercise, a low meat diet becomes increasingly attractive as an option for long term health.
At the very least, try a vegetarian diet for two to four weeks, and see if you notice any appreciable changes in how you feel. If you notice a difference, try and find ways to incorporate more non-meat options into your diet. The goal isn't to "convert" you to becoming a vegetarian, but to make you aware of the health benefits and changes in how your body feels and pave the avenue for long term health.