Several muscle groups in your back and torso...even your rear end...make up what are known as your "core." This region is the center of your body, and naturally, it helps to stabilize your body in just about any form of exercise you do. Strengthening your core will increase your overall strength, improve your posture, and help you with your balance. You can see why this would be an important muscle group to work.
Fortunately, many exercises that you do will naturally work your core. However, special attention should be given to make sure your core is nice and strong. Here is a list of some popular workout routines to strengthen your core (there are many variations of these exercises, but I will give only the most basic forms as I understand them):
1. Traditional sit ups (slight risk of injury). If you are going to do this, lock your feet under something heavy like your couch. Place your hands behind your head. With knees bent, keeping your back and neck as straight as possible, lift your torso up to your knees. The tendency is to try to pull your neck instead of lifting your whole torso. This creates a lot of stress to your spine which could be a problem. I have heard that you should visualize something like an apple between your chin and your chest to help you lift your torso instead of your neck (I suppose that works). This exercise uses many more muscles than just your abs, and although there is a risk involved, I recommend it as long as you work on quality before quantity. Taking it slow and working on your form will make it a safer and more effective workout for you. (Start with about 10-20 repetitions)
2. Crunches. These are a bit safer than the sit ups, but they focus primarily on your abs. Start just as you did with sit ups only don't lift all the way up. Focus on tightening your abs each time you lift and then return to start position. You should be able to do quite a bit more of these than sit ups, but they are very affective in strengthening your abs (only remember that with this exercise you are leaving out many other important core muscles, like your back and waist. (You can probably start with 30 repetitions)
3. Leg lifts. The basic form of leg lifts is to lay flat on your back, arms on the floor along your sides. Lift legs up a couple feet, hold for a second, and then return to floor and repeat (10-20 repetitions would be a great start)
4. Planks. With this exercise, you basically place your forearms on the ground and, lifting your rear to keep your spine straight as a board (or "plank") from your neck to your feet, hold that position for as long as you can. (30 seconds to a minute would be a great start)
5. Bear Crawls. This is an exercise I would probably recommend the most for core strength. I guess it can be a little technical, but it uses just about all of your core muscles. You don't have to be overly worried about form. Basically, just crawl on your hands and feet, going forward a few yards and then backwards. Ideally, you want to keep your back as straight as you can and your knees bent close to the ground. It may take some practice, but it is very effective...and probably more fun than the other choices.