(Remember: I'm not a doctor, so all standard disclaimers apply. Proceed at your own risk!)
It occurred to me that some of you who haven't worked out much might wonder what the big deal is about warming up before and cooling down after a workout. So here's a short, non-technical reason for each.
The idea behind working out is that you increase your body's ability to handle extra stress by gently pushing it beyond its current ability. (That word gently is important. Pushing in a "non-gentle" way is what causes most injuries.) In order to give your body the best chance of handling that extra stress without getting hurt, we want it to be as flexible as possible when we make the effort... and muscles are most flexible when they have been warmed up. That's just another way of saying we have increased the blood flow through the muscles, and we do it by doing some really gentle exercise that doesn't push them beyond their current ability. Simple enough, eh?
Warming up applies to any type of exercise. Cooling down applies primarily to aerobic exercise.
You might wonder how aerobic exercise helps your heart. Doesn't it just make your heart beat faster, like any other exercise?
In a word, no. Aerobic exercise does make your heart beat faster, for a prolonged period of time, but there's more to it than that. In aerobic exercise, the large muscles of your body (like your legs) take over part of the pumping work that your heart normally does on its own. As a result, your heart can beat faster without working too much harder. And we limit our pulse rate because we want to keep that faster rate at a safe level. That's the key to how aerobics work.
However, if your legs are doing some of your heart's work and you just stop, your heart suddenly has to start doing all the pumping work again. If it does it successfully, it puts an extra strain on your heart that we don't want. Many times, your heart won't be able to do the job; if you've ever seen someone just stop after a long run and then faint, that's why. (You see this sometimes at the Olympics, but it happens with weekend athletes as well.)
The way you prevent this is by cooling down, which simply means you slow down but don't stop moving. If you've been working out within your limits, slowing down to a walk for a few minutes is usually all you have to do to cool down. (Some people like to swing their arms a little bit as well.)
Personally, I usually incorporate the warmup into the first minutes of my aerobic workout by walking briskly for two or three minutes; and then I cool down by slowing to a walk for about five minutes or so at the end. Sometimes I run a little less, then spend 20 minutes walking at the end and just consider that part of my workout. (For some of you, alternating walking with brief periods of jogging will be a good way to get some of the benefits of running without having to run continuously.)
And that's the why and how of warming up and cooling down. It's simple to incorporate them into your workout, and now you know why you should and a simple way to do it.