I worry that my fellow commenters may be glossing over the really important aspects of meditation. Here's my understanding:

There are many different types of and approaches to meditation, but the essence of all of them is mindfulness. The concept is of being "present"-- not living in the past, or the future, but experiencing what is happening right here, right now.

This is a subtly but importantly different idea from that of ignoring or quieting your thinking. You are your thoughts; you can't ignore them. (Don't think of an elephant.) However, your thoughts are not all that is happening to you, and most of the time your thoughts serve only to distract you from what is actually going on.

So, for example, one of the most basic forms of meditation is simply to be aware of your breathing. You are breathing one hundred percent of the time, but you are almost never aware of the fact that you are. So take one moment, right now, close your eyes, and breathe. Feel how your chest expands and contracts. Feel how the air flows through your nostrils.

Thoughts will occur, and it's important that you don't push them away, or try to ignore them, because you can't-- even very experienced meditators cannot. Instead, recognize that thought is occurring, be aware of it, but don't dwell on it. Let the thought exist, notice that it is happening, and let it drift away as you return your attention to your breathing.

You will find your mind drifting to more and more intrusive thoughts: The things you're stressed about, worried about, upset about. This is not wrong; it is what is happening to you right now. Be aware of it, let it pass into you just like the air through your nostrils, and let it pass out as you return to breathing.

If you really attempt this, you will notice at least a brief moment where you feel suddenly, absolutely aware of who you are, where you are, what it is that you are experiencing right now in a way that you may never have felt before. If you've never meditated before, it is very likely that you'll be terribly upset by this awareness of all of the unpleasant thoughts and feelings that you've been ignoring, trying to push under the surface and bury inside yourself. Remember that it is okay to cry. Let the emotions happen, just the way they are. You can't make them go away. But they don't control you; once you've felt them, it's okay to return your attention to your breathing.

I hope I don't make this sound easy-- it's incredibly difficult. But over time, with practice, you'll find this state of presence and awareness easier and easier to achieve, and that it's no longer necessary to close your eyes or be still to feel this. You'll be able to experience exactly what it feels like to walk down the street, or to eat a meal, in a way you never have before. You'll find yourself no longer being ruled by the emotions you've tried to suppress for so long. You will be angered, or saddened, or terrified, and you will be able to let those feelings happen inside of you, really experience them, and continue with what you are doing.

That is meditation.

(My references: My mother is a clinical psychiatrist who is trained in mindfulness-based cognitive therapy, which is, believe it or not, a clinically validated treatment for many serious mental illnesses, but especially depression and anxiety. I've used these techniques myself to overcome major recurring depression.)