Welcome to the board. There are a number of ways to set up a home studio. The variables include: what type of music you are recording (style, band vs solo, acoustic vs electric, MIDI, etc.), what you already have, and how much you are willing to spend. If you already have a decent computer with a big hard drive and plenty of RAM, you've likely covered your biggest single expense.
The necessities:
1) a computer interface: your computer probably already has a soundcard, but most stock soundcards are not good for recording. M-Audio and others make good full duplex cards with inputs for recording audio, MIDI, etc. Some of them have an external box with preamps and stuff. You can also buy an external interface that connects via USB or FireWire. There are pros and cons to each. Check the Computer Recording forum for more info. Prices $150-$$$$.
2) Mixer/Preamps: If the soundcard does not have a preamp (or if they aren't very good), you will need external preamps or a small mixer. If you are recording one instrument/voice at a time, preamps may be all you need. If you are/will record multiple instruments (or a drumset), it may be adviseable to get a mixer with 8-12 channels. Again, prices and quality vary widely.
3) Microphones - You will want a few for different applications. I would get at least one Large Diaphragm Condensor mic for vocals, a pair of Small Diaphragm Condensors for recording acoustic instruments (and drum overheads), and at least one decent dynamic (SM57, or similar).
4) Recording Software - The Recording Industry standard is ProTools, which comes (in a limited version) with some of the computer interfaces (such as Mbox). You can save some money by going with Cubase or Sonar, which are excellent programs in their own right, and may be just as good, if not better, for the home recordist. Most of these programs come with built-in effects plugins (reverb, compression, etc.) and there are plenty more available either free or purchased. Less expensive programs include Powertracks and Tracktion. Price $50-$$$$
5) Monitors - You will need some nice speakers to listen to your mixes. These will be important to making good mixing decisions. Price $300-$$$$.
Those are the basics. There is a wealth of info on this board regarding which equipment will suit your needs (the search function is great). Recording at home is fun, but can get very pricey, as you will continuously realize the limitations of your current equipment and want to upgrade. Good luck.
