A siphon is a thingy that makes water go up. Basically, it looks something like this:
Atmospheric pressure pushes against the water, shoving it up the tube. When the water gets high enough to hop over the little bend at the top, it gets pulled down by gravity the rest of the way. But when that water gets pulled down by gravity, it leaves a vacuum behind. So there's a pressure differential, and that gives you a net force that shoves more water up the tube. Of course, for this to work, the whole tube has to be full of water or else the air in the tube will push right back and nothing interesting will happen.
If you've ever seen a barometer, you might be familiar with the pressure thing, since that's the whole idea behind a barometer. You've got a setup like this:
The atmospheric pressure shoves down on the water, forcing some of it up the barometer, and you can tell how strong the pressure is by looking at how far up the tube the water goes.
Well, a toilet bowl is set up to take advantage of this stuff.
See, when that pipe is full of water, gravity pulls the water down, right? But when gravity pulls that water away, it creates a pressure differential, allowing the atmospheric pressure outside the bowl to shove more water into the pipe. This push-pull siphon action is what sucks the flush down into the sewers. You know the gurgle-gurgle noise a toilet makes after you flush it? That's the air bubbling back into the pipe once the water is gone.
So flushing a toilet is basically just dumping a whole bunch of water into it all at once. And in fact, you could manually flush a toilet if you really wanted to for some reason. All you'd have to do is take a big ol' bucket of water and dump it in there and, if it was enough water, it would flush, just the same as if you pulled the lever.
Okay, so what is the rest of the toilet for? It's got that whole tank thing with weird floats and pulleys in it. They've got to be used for something, right? Well, yeah. That's where the water to flush the toilet comes from. If you take the top off of a tank, you'll probably notice that it's full of water. That water is the water that goes into the toilet bowl when you flush it.
When you pull the handle, it tugs on a chain that lifts a valve at the bottom of the tank. With the valve open, the water in the tank pours out around the edges of the bowl and through the jet-thingies around the rim, rapidly filling the bowl with water and initiating the flushiness. You can see the water drain out of the tank if you take the lid off when you flush.
You'll probably also notice that there's a float and some tubes and a nozzle thing in the tank, too. Those are for refilling the tank once you've flushed the water out of it.
When the tank empties, the float goes down (because it's a float, and it was floating on the surface of the water). The filler valve is hooked up to the float and notices it dropping. "Hey," says the filler valve, "The float just dropped. That must mean the tank is empty. I'd better refill it." So the filler valve pours some water into the tank and some into the bowl. As the tank fills up, the float rises, and when it gets to the top, the filler valve notices and says, "Aha! The float is back! I'll stop now." And the water stops.
That extra tube is there in case something goes wrong and the tank starts to overflow. The extra water will go down the overflow tube and into the bowl, and then down through the siphon pipe, instead of overflowing out of the tank and all over the floor.
Sometimes a flush goes wrong and your toilet gets clogged. This is a terrible thing. Usually it happens when something gets flushed that's too big to flush, like a large wad of toilet paper, or a kitten. Contrary to popular belief, it is not caused by evil toilet-clogging gremlins, although if it were, that would totally make sense.
When a toilet gets clogged, that means it's time to bust out the handy-dandy plunger.
Toilet plungers have a pretty simple strategy. The plungey-thing at the bottom has air in it. So what you do is you take the plunger and stick it over the hole at the bottom of the toilet bowl and push it down to turn the rubber cup inside-out. When you do this, it shoves the air down the pipe. And then when you pull it back right-side-out again, it sucks up a mix of air and water.
Basically, it churns everything up. And when everything gets all churned up, the force of the water and air swishing around helps to dislodge the kitten (or other debris) that's clogging the pipe. This could take several tries, of course.
Anyway, that's pretty much how your standard toilet works.