'Dodging' and 'burning' derive from traditional photography in the dark room.
If part of a photo was too dark the photographer would hold a hand under the enlarger to stop the light the light reaching that part of the image for a few seconds. That part of the image would then print lighter than the rest. This was known as dodging.
The converse was to use a piece of card with a hole in it to allow more light to fall on part of an image which was too light - or to 'burn' that part of the image.
We have already seen how we can lighten or darken a whole image. And we have seen how we can use a brush to paint over parts of an image. We can now combine the two to do just what traditional photographers did.
The dodge tool is at the bottom of the toolbar. You might need to right click to find it.
Select a reasonable brush size and 'paint' over your image. You will see it gets lighter in just the parts you have painted. Normally you'll use a fairly large soft brush with low opacity to get the effect you want.
Don't overdo it. If the area you are trying to lighten has no detail in it (i.e. is completely black) then dodging won't help.
Just like dodging, but makes things darker. Again if the area has no detail ('is burnt out') you cannot create any by burning.
This is one that wasn't available to traditional photographers but works in the same way as dodging and burning. You can use it to increase of decrease the saturation of the colours.
Download the left hand picture below (double click on it to get the full size version).
Use a large (27 pixel) soft brush with opacity about 10% to dodge the shadows around the eyes and neck.
Use a small (5 pixel) brush to lighten the whites of the eyes (you'll need to zoom in to do this). At the same time you might want to improve the catch-lights. Use a very small, 3 pixel hard brush to paint (not dodge) a white spot. Use opacity about 20%, otherwise it will look overdone.
Burn in the path behind the girl with a 45 pixel soft brush. Note that you can't really improve this picture because there is no detail there anyway.
Finally use the sponge tool with a large brush to desaturate the background a bit.
My final result is the picture on the right but you can produce whatever effect you prefer.