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I'm building this tutorial for everybody, from the novice to the knowledgeable. Perspective is something that gives a lot of artists a LOT of trouble, mostly because these artists are unaware of the simple math behind the concept. I'll try my best to keep it clear and simple. 

My father's an architecture designer. I have him to thank for my artistic talent, as well as for being my inspiration to draw ever since I was little. Dad taught me about perspective a long time ago when I was about 4 or 5 years old, and back when I was working for his company, he trained me in the more mathematical aspects of perspective for me to do perspective renderings of the houses he designed.

First of all, I want to start by saying that perspective is not difficult once you grasp the basic concept. In order to help you understand it better, nothing beats looking at examples and practicing. 

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Perspective is drawing the illusion of a third dimension. Before I get into that concept, though, I'd like you to visualise things around you. Stand in front of a chair, facing straight ahead, and crouch down. Look at the chair.

The chair in the example here is straight, with a square seat. This is to simplify things here, however, you'll notice in your real chair what I'll indicate below:

This is a classic one point perspective. You will notice that the sides of the chair go back in a trapezoid shape, and if you could trace lines going along the sides of your chair, they would meet in the middle, at your eye level. Same thing goes with the legs of the chair, they are alined going back towards that one middle point at your eye level. 

Your eye level is called Horizon Line. All the flat, horizontal surfaces whose edges go away from you head towards that vanishing point that is always on the horizon line. All the vertical lines remain vertical and parallel, and the horizontal edges that athat face you are also parallel.

In truth, these vertical and horizontal lines are curved around you -- but the curve is so subtle that their vanishing points are immensely far from where you stand. For example, say you're standing in front of a sky scraper. If you look at it straight ahead, the windows look parallel in front of you. If you look up, however, you'll see that the building is getting smaller as it goes up, to form a sort of trapezoid shape. What makes this different? The placement of the horizon line -- i.e., from where you're looking. I'll get into that later.

Now, say you're standing just a little to the side of that chair.

The angle of the chair, in your point-of-view, has changed. The square seat becomes skewed. This becomes a 2 point perspective.

The horizon line now has 2 points on it. One of them is for the edges of the chair that go towards you (forward) and the other is for the perpendicular edges (to the side). The vertical edges still stay parallel in this kind of perspective.

"Okay, this is all nice," you'll say, "but what do I do with characters now? How do I make it look like my character's actually sitting in this chair?"

When you break down a character into its basic shapes before you draw it, keep perspective in mind. The next drawing shows a character sitting simply, straight and basic, in a chair. I did it in stick figure mode to help you understand what this means.

Use the same vanishing points that you used to build the chair to place the shoulders, the hips, the knees, the ankles, the hands and elbows. Use them to orient the thighs and the feet. This character's thighs are perfectly parallel, yet look at how they form a skewed trapezoid shape. Look at how the feet seem to be heading in different directions, yet your brain knows they are pointed straight ahead in comparison to the chair. Look at the angle at which the shoulders are oriented compared to the feet. Yet you know for certain that this person is sitting straight in his chair.

Another example in using perspective for characters is as follows: Say you want to draw two people at opposite corners of a street. You'll draw the street using a horizon line and vanishing points, right? Well, use these vanishing points to establish your characters in the scene, and to show their size relation. Below I drew the same character close to the camera, and further away. I drew a chair mext to them to show a size relation. Both characters are the same size! One is just much further than the other.

About vanishing points: they don't have to always be on the paper. Sometimes they'll go out of the paper, like the example above. In order to remedy that, you can either tape an extra sheet to extend your horizon line far enough to find your missing vanishing point, or you can trace a grid of indication. What I mean by that is to take a reference line perpendicular to the horizon line, towards the egde of your paper, and gradate it (i.e. trace a notch at every half-inch, say). Closer to the other vanishing point, trace another perpendicular guide line that will be notched at every inch, for example. Then play connect-the-dots. These fanning lines can serve as guides for your "to the side" lines.

Just a note about how perspective can be applied when drawing normal, just-standing-there characters: if you look at someone standing some length away from you, you will notice that their feet aren't on a flat, horizontal line. Neither are their shoulders. The foot that is closer to you will appear lower if looked at in a 2D view, and the shoulder closer may look higher than the other one. Using that in your drawings makes a BIG difference in losing stiffness in a drawn pose, as well as giving the image depth. Just look at the character above. Look at his feet, look at his shoulders. 

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Remember when I mentionned looking up the skyscraper? Think also about looking down from a balcony. There is a third vanishing point that controls the vertical lines. Since they are going away from you, they will, at infinity, converge, at least in your eye. Downshots are easier than upshots  in my opinion -- we are more used to seeing people from above than from below. 

A downshot happens when you look below the horizon line. You still have the two vanishing points on the horizon line as explaned above (Remember math and geometry? Think of them as your X and Z axis) and you add, low, perpendicular to your horizon and smack down in the middle of the two othe vanishing points, your Y axis, 3rd vanishing point. 

As indicated in the drawing, drawing a "floor" grid helps to position your characters on the ground that is below the camera (think ceramic tiles, for example). My third vanishing point is way below my paper in this drawing, but you can still tell that there's an angle in the way the characters are standing, depending how far to the left or right of that central Y axis they are. Downshots work especially well with making small characters look even cuter. 

All the lines that are normally "horizontal" (shoulders, knees, hips, ankles...) will always be oriented to the same vanishing point on the horizon line if the character is standing straight, like examples 1, 2, 4 and 5.  Example 3 is slightly turned, so his shoulders are aligned to another vanishing point on the horizon line, slightly away from the vanishing point that controls his lower body.

I find upshots trickier, but I still like them. The camera's POV is above the horizon line, and the third vanishing point is high up there. 

This drawing is skewed, though. The Y axis is crooked, but you still get the idea. Horizon line and Y axis should always be perpendicular. As you can see, the "horizontal" lines of the characters' bodies aling with the vanishing points. All the "vertical" lines aling with the third point that's placed above. A better example of this is shown below.

Q: Where do I place my horizon line and my vanishing points?
A: There is no preset rule. Mostly you'll figure it out with trial and error. However, you will notice that the closer you put your vanishing points to one another, the more skewed your drawing may look, especially if you draw large. Put them as far away from each other as necessary. Tape several sheets of paper to the sides of your drawing if you need to.
The horizon line is your eye level. If you're standing straight upright and you look straight ahead at, say, a road, the horizon line is exactly even with your eye level. That's why the horizon is at different heights for different people. The horizon line lowers when you sit in your chair. Now, say you're on a balcony and you're looking down. The horizon line is at your eye level still -- the eye level of when you're looking straight ahead! It doesn't lower with your gaze. Same if you'ree playing the romeo to your juliet. The horizon line is at your horizontal eye level.
Lie down on the floor and look straight forward. You get a cat or a dog's POV. The horizon line is really low and everything looks like it's angled upwards. Then climb on a chair, or even on a table. Doodle what you see, and find your horizon line and your vanishing points. Try it, it's fun!

Q: How do you remember where your vanishing point is? I keep putting my pencil on it as a guide and now it's a big blob...
A: A trick of the trade is to use thumbtacks and either masking tape of Scotch Magic Tape (the kind in the blue box). All you have to do it tape the thumbtack upside down on your vanishing point (pointy end up) and use it as a stopper for your ruler. That way it will be easier to keep the same vanishing point, and you won't have to refer to two points all the time to trace your lines.

Q: What about lines on your character that aren't parallel to the perspective lines (like lines that show muscles)?
A: The use of perspective lines is there to help you make the basic construction of your character, you have to flesh him/her out afterwards. If you mean how to decide the angle of lines like the shoulders or the hips, my take is to draw the "straight" horizontal guide, then give that line an angle. I'm not sure I really understand what you mean, though. Using the perspective lines do help when finding reference points, say for example the lowest part of the breasts, or the highest point of the bicep.

Q: What if a character's pointing diagonally against all the lines?
A: All the parallel horizontal lines have the same vanishing point. It will invariably be on your horizon line. If the body is twisted, say, the hips, knees and ankles would roughly point to the same vanishing point, and the shoulders and ribcage would point to another vanishing point on the horizon line. Now, if you're thinking of sloping lines, the mathematical way of figuring that out (in 2 point perspective) is to extend a vertical line from your vanishing point and make measured, even notches along it, so these become vanishing points for sloping lines. That is used in drawing the roof of a house, for example. If you want to make sure that both ends of the roof have the same pitch, figure out the angle of one side of the roof, then extend it onto your vertical line, and mark where the lines meet. From that cross point, all sloping lines parallel to this one will refer to that same vanishing point.
VANISHING POINTS EVERYWHERE!