There are a lot of reasons why you may want to scan your artwork. If you want to create a run of prints from a single original physical work, it will need to be scanned. Scanning is also a great way to archive your work in case of a disaster (like a fire).
And finally, if you want to share your artwork online, it is a lot easier to scan it versus trying to get a decent photograph. Between having to control the lighting, to rendering the colors properly, and making sure that the camera faces the artwork exactly head-on, a scanner is definitely the way to go if you can.
If you are planning to create prints, you will need to scan in your work at 300 dpi or 300 ppi. This is the industry standard for prints that will look just as good as the original. And trust me, it is FAR preferable to get the scanner to do 300 ppi versus scanning it in at whatever resolution, then changing the image's resolution in Photoshop.
The terms "dpi" and "ppi" are used interchangeably. "dpi" stands for "dots per inch" and is specific to printing with physical ink. If you have ever looked at a newspaper photograph under a microscope, you have seen all the dots that make up the picture. Pack 300 of those dots into an inch, and you have something that looks seamless to the human eye. "ppi" is the same concept, although it stands for "pixels per inch" and specifically references the digital side of the equation.
Conveniently, scanning something at 300 ppi also means that the scanned image will be the same size as the original. If you take an 8x10 painting and scan it in at 300 ppi, then print out the image file on your printer, you will get an 8x10 image.
However, if you scan it in at (say) 200 ppi, the resulting image file will be smaller (something like 6x4). Obviously if you want to create prints the same size as your original (which you probably do) then you want to use 300 ppi.
You may need to fiddle with your scanner's driver settings to get it to scan at 300 ppi. In Windows 7 you can go to Start -> Devices and Printers, then double-click on your scanner and poke around until you find the resolution settings.
If you don't have your own scanner, you can take your artwork to a place like Kinko's to have it scanned. Be sure to confirm that they will be scanning it at 300 ppi before you hand over your cash!