So now your eggs have hatched and there are little wet birds in the incubator...
Leave them in there a few hours. Let them dry out, if you take them out while they are wet, they can chill and die.
Make sure your brooder is set up. Please don't use woodchips. The chicks are capable of eatting the smaller pieces, that would cause their intestines to get blocked and they can die. Don't use anything slick, like newspaper. They can't get a good footing and their legs can get deformed... their bones aren't hardened yet.
Make sure you have a heat lamp and that the temp. is around 100 F (every week after the temperture goes down by 5 F until you hit the room temperture (or outside temp. if you got them outside)). Give them a waterer and feeder. Make sure their water source is drown-proof. You don't want them to be able to get in and wet... they can chill or drown. It is easier to just buy a "chick waterer" - you can always wash and reuse it.

Feed: Use a quality chick starter... you can even find it organic. You will use this feed for a couple months so don't be afraid to save some money and get the 50# bag.
How to tell if your chicks are too hot or cold?
If your chicks are all running about (after the first day of rest, of course), then you have the temperture about right.
If they are huddled together in one big ball, then you have it too cold.
And if they are not at all "congrigating" and panting a bit, then it is too hot.
Just put the light closer or farther away depending on your situation, and check back to see if you got it right.
I always give them a "warm end" and a "cooler end" in the brooder so they can decide. I never use thermometers (and never lost one chick), as the first hatch I ever had I tried, but after 2 days it was impossible to read... they kept knocking it down and pooping on it.

Pasty butts:
It happens. Chicks have fluffy butts (or vents, really) and when they go, it can get the fluff dirty, then it dries and builds up... this can lead to a mass of chick poop blocking the chick from being able to go again... which, you guessed it, can lead to death.
Just take something like a q-tip and a pick and try to get the area clear again. If you have a bigger, harder mass to deal with, it may be needed to soak the little one's bottom in some warm water to soften it. Once again be careful to not soak the chick... it can chill and die.
Once you got the poop soft enough, you can get the vent back open with a q-tip and/or a toothpick.
You should check back often after this... it tends to be a problem again sometimes. Good news is, it won't last forever. Once they feather out, it isn't a issue.
Check on your chicks every few hours (do I really need to say that? You'll be in there all the time playing with them, right?). Make sure everyone is walking right, eatting and drinking (the first 3 days they can live off the yolk from the egg - so don't panic if they aren't scarfing down durring that time), and make sure no one seems "different".
All this is great, but what is a brooder?
A brooder is just poultry-speak for a containment unit that houses chicks and keeps them at the right temperture while they grow for the first 7-8 weeks of life. It can be a box, a rubbermaid container, or one of the fancy metal stackable ones the professionals use. I happen to use rubber cattle trophs. It's up to you. Just make sure it is draft proof. After a couple weeks (even sooner with some breeds like Delawares) they will learn that they can fly. (and chicks CAN FLY) You need to be aware and prepared for this stage. It is my favorite age for chicks... like the terrible twos! They learned they can fly and fly they will!
If you have something happen or have a chick that has a issue, BYC is a great resource, but you can always email me. If I can't help you I can lead you in the right direction.