There are many who believe that Burn and Dodge are the devil,and should not be used because they make artwork (and edits!) look bad and cheap, but this need not be the case. Used properly, you barely even realize that they are even there in the finished product, since it looks so natural. I stand by my statement that they are the best way to make something look very shiny, like silk, metal, or hair, but only if used sparingly. It is very easy to overexpose something with these tools - especially with Burn - and they should not ever be used by themselves.
How these tools work
Both the Burn and Dodge tools have three settings for "Range," labeled "Highlights," "Midtones," and "Shadows." What this means is which parts of the colors get brightened (for dodge) and darkened (for burn). "Shadows" has a greater effect on darker colors than lighter ones, and "Highlights" has a greater effect on lighter colors. "Midtones" affects all shades equally.


Above are two examples of Burn and Dodge on all three settings. In both images, "Shadows" is at the top, and "Highlights" is at the bottom.
It is important to note that the "Highlights" and "Shadows" settings have a significant effect on the saturation of the image. Burn on "shadows" and Dodge on "highlights" will increase saturation, while Burn on "highlights" and Dodge on "shadows" will decrease it. Because of this, you can use these tools to cancel each other out, if they're on the same range setting and exposure level.

Above are examples of how Burn and Dodge cancel each other out. At the left is Midtones, at the right is Highlights, and in the middle is Shadows. As you can see, the overlapping areas of the strokes are the same color as the surrounding area. Below that is an example of how putting Dodge highlights on top of Burn shadows can make the saturation doubly bright.
Exposure
This is what you do to control these tools, and not let them ruin your work. Remember that the Range settings can and will influence how much exposure you want to set the tools at. Midtones are not normally a problem, and Dodge shadows and Burn highlights are rarely used because they make things look grayer, but Dodge highlights and Burn shadows are useful - even if they are tricky to control.

As you can see, lower exposure has less effect on all the different shades, but the effect is amplified immensely by the darkness or lightness of the basic shade.
For Burn shadows, the darker the object, the lower you want your exposure - but for lighter objects, you'll need to turn the exposure way up. As shown, the low-exposure Burn is perfectly visible on the dark shade, but you can barely see it on the middle shade and not at all on the light. Turning it up to a point where it looks nice on the light shade makes it too dark on the dark shade, so you must always adjust your Burn for color.
Dodge is a bit less finicky most of the time, but you must take great care with very light shades, as it can easily wash them straight out to white.
Using these tools effectively
All right, so this is what you do and don't do. First of all, DO NOT shade and highlight the whole thing with nothing but Dodge and Burn - that'll make it look like this:

That little pink dress was shaded and highlighted using nothing but Burn and Dodge. See how it looks like cheap plastic instead of silk, like it's supposed to? There's no crispness to it, and the whole thing looks messy and half-assed. This is how to make better use of the Burn and Dodge tools.

Above is the first step toward good burning and dodging: Good brushwork. There are only three colors here - a base midtone, some shadows, and a few little highlights. It looks a bit gray and flat, but that doesn't matter, because the saturation that the Burn and Dodge tools add will really make it pop. However, this step isn't a total pitfall avoider. Observe:

It's slightly better than the first example, but the highlights and shadows are overexposed, bringing it right back to Cheap Plastic! While overexposure like this is okay on metallic objects like armor, it's terrible on cloth and hair. You can avoid this by using big, soft brushes and only a very few judicious strokes, and not trying to get too much detail out of Burn and Dodge. Detail is what the paintbrush is for, and Burn and Dodge are the worst tools available if you want to make something more distinct. Do not burn every single shadow one at a time with a small brush - burn them all at once with a big brush! It looks much better.

Above is an example of Burn and Dodge done right: All the details and sharpness come from the preliminary brushwork, and the shadows were put in at low exposure with a great big fluffy brush. The highlight brush was slightly smaller than the shadow brush because the highlights themselves were smaller, but it was still about ten pixels wide. Again, this brush was soft and fluffy, and to put in the shadows and highlights I only used about two strokes each.
There is a lot of Undoing when you're choosing the right exposure. If it's too low, don't just go over it twice - Undo, and set the exposure higher to try again. Do the same if the exposure is too high.
I hope you learned something from this tutorial, and happy shading!