Things to know- I colour in Photoshop, currently CS3, but I think versions 5 and up work pretty much the same way, with a few differences in the names and layout of the tools. I'm positive you can do the same thing using other image editing programs, but don't ask me how...cause I have no idea!
First off- When you first scan, your lineart is on a layer called Background. It's locked and you can't do much with it other than draw over and wreck it! See the little padlock icon in the Layer window? You can "lock" certain aspects of a layer by clicking those. First one is transparency: locked means no matter how much you colour, you'll never go outside the lines. It also means that if you have something semi-opaque, you'll never be able to make it less see-through. Second and third aren't as important right now. The last is Lock All, meaning you can't change anything. A background layer is automatically locked, but you can still draw over because Photoshop doesn't understand that the black lines are supposed to be separate from the white paper. Soooo....

1- Double-click the layer in the layer window. A window will pop up, click yes and your BG layer will now be "Layer 0" and you can do stuff with it.

2- In the Channel window, click the little circle icon, the one that looks like an exploding bubble (first on the left usually). This selects all the white in the image (so make sure the part of your image that you want to get rid of is white and not off-white). Hit the "delete" key. Now you've got only lines on a transparent background. (Note: if you drew in pencil or any other colour than black, doing this will eat away at your lines, that's normal, but you can fix it later!)

3- Now like I said, unless you used black ink, deleting the white in the image will leave with a portion of the line colour. So to fix that: Back in the layer window, there are a bunch of little icons at the top beside the word "Lock:". Select the checkered square. (mouse over it and it says "lock transparent pixels" or something like that. This means now you can only change the pixels that already exist on the layer.

4- Fill with the colour of your choice. (I used black, see how the lines have gotten thicker on the left image? That's how they're supposed to be, and how they looked when I first scanned the image in. If I had done this in pencil, I'd need to duplicate my line layer once or twice to make up for losing some of the line colour when I took out the white)

5- Create a layer below this one and fill it with white or whatever background colour you want for your image.
When you get to the part where you want to colour your lines (this is something I only do at the very end of colouring) all you have to do is keep transparency locked and draw on the lines in the colour you want!) With the lines on their own layer, you can do all sorts of crazy stuff with them. Like making them a horrible neon green!

6- Or not! Black ended up being fine for this drawing...

Hopefully this made at least a little sense!