@ r0ckstarlette D&B is all about "adjusting the tonal interrelationships" in an area by simulating an increase or decrease of exposure, just for that area.
dodgingIf a sign in shadow is a little too dark to read, and you want it to be read, dodge just the sign, and you will lighten it in comparison to the areas you didn't dodge.
burningHas someone's face in a group shot got caught in bright sunlight (but the rest are normally exposed)? Burn the bright face to darken it.
Of course, with tools like Photoshop and others, you get a lot of control in applying those tools. Like being able just to burn the highlights, or just to dodge the shadows.
So in brief, that's it.
Of course, like anything, it can be used very creatively, if you know that using it in specific ways gives specific, predictable results.
making texture tactileFor example, I use it a lot in shots where I'm concentrating on texture as part of the overall feel. Remember, I'm really just simulating what the eye is capable of if it were there (which mine is and was!).
the eyes have itWe don't notice it, but as we look around, our eyes are continually adjusting exposure. Even in bright daylight, if we focus on a small patch of shadow, our irises open up a bit to let in more light - and we see the contrast between light and shade within the shadowy area - look away and the irises close down again in reaction to the brightness they're now looking into.
Still cameras can't do that, so we resort to d&b.
exampleHave a look at this shot where I've used creative dodging and burning extensively in the grass - especially in the f/g. I concentrated on dodging just the highlights and when the mid-tones got too close to the highlights, I burned just the mid-tones back down again.
I did all this on a new overlay layer, filled with overlay-neutral grey. I dodged and burned on that layer and then adjusted the opacity to fine tune things. (and deleted the layer and tried again a few times till I was happy - you gotta 'speriment!)
The key to it as mentioned by others, is a small soft brush, a low strength and long sweeping movements. Never burn the highlights, and never dodge the shadows (and remember that the purpose of having rules, in the words of author Terry Pratchett, is so that you think before you break them).
Also, I'm not an expert (learning, learning, every day a schoolday!) but the above summarises what I've learned so far.
Everyone else, please free to correct me or comment. If you're interested in bw, use your flickr id to join my website via google friend connect - www.highton-ridley.co.uk
Hope that helps,