Well, pretty much all you need is:
* A PC
* A good quality sound card
* Microphone preamp
* Multitrack recording software
Plus eventual microphones, direct boxes, cables and of course, instruments.
PC
For optimum performance:
* The PC OS should be run as clean as possible, by having a dual boot system or simply a dedicated DAW PC.
* It should have 7200 RPM or faster HDD's.
* PIV is supposed to be a no-no. I've heard nothing but bad words about Pentium IV processors when used in DAW configurations, everyone seems to prefer AMD CPU's or even Pentium III.
* At least 128 MB RAM. Preferably as much as you can afford/the OS can handle.
The
sound card should be 24 bit. 24 bit resolution allows you to alter the wave files more, by apply more effects and whatnot, before the sound quality starts to degrade. All pro cards sold today that I know of, are 24-bit, but a mid-level card such as the Soundblaster Audigy is not and therefore isn't an ideal recording card.
Latency is almost always an issue with DAWs and sound cards... measured in milliseconds, and something that should be as low as possible. To achieve low latency levels, you need a good sound card with good drivers. Latency is an issue when doing overdubs - even if you think you played it exactly in time, as tight as a virgin's whatever, it sounds sloppy when played back. It wasn't you, it was the sound card.
If you record with some sort of "direct monitoring" system, latency is all of a sudden no longer an issue when recording. I don't know exactly what it is, and I don't know how it works, but I have it and it's all good.
A sound card has a certain number of inputs. 2-channel cards are the most common, with 2 mono inputs. There are certain cards with more inputs, and there are cards which feature a so called breakout box, which can give you more channels. E.g. M-Audio Delta 1010, Aardvark Q10 and STAudio C-Port are all 10-channel devices - capable of recording 10 tracks at once. If you want to do a live recording of an entire band, this is what you should look at.
I'm sure JMX can tell you more on the sound card bit.
Microphone preamps
Come in a multitude of shapes, sizes and prices, with different features and sonic character. It's a jungle out there, and you have a number to choose from even in the bottom price league nowadays. Mixers contain mic pres, so if you choose a 10-channel soundcard, using the pres on a mixer might be more convenient than having 10 separate small separate mic pres.
Software
Also a lot to choose from. If you buy the software, prices can vary greatly. If you use pirated stuff, nevermind that. Download demos and see what you like and are willing to buy (or not). I'll just give you some names to do searches on... N-Track, Cool Edit, Cakewalk Pro Audio, Cakewalk SONAR, Emagic Logic, SAW, Cubase, Nuendo... did I get them all?
To record stuff
To record guitar, sticking a microphone in front of the guitar cabinet is the most preferred method. How you place the microphone makes a whole world of difference though, so it will take some experimenting to get it to sound the way you want to on tape.
To record bass, using a direct box or preamp with a DI is the most common way. A DI circuit lowers the impedance of the signal, which gives a cleaner tone and more headroom available to the mic pre you run the signal to.
You can also mic a cabinet, or do both. It's all in which you find suits you best in terms of sound comfort.
Recording vocals is pretty straightforward; sing into the microphone like you always do.
Recording drums is probably one of the toughest things to do with a good result. Not only do you need a number of different microphones (4 is usually a preferred minimum - kick drum, snare and two overhead mics), but the editing and mixing processes can be tricky or just tedious for a newbie, since every part of the kit bleeds into all mics. Lots of noise gating/wave editing and probably also EQ magic needed to get it good.
You know what? I'll stop there. I wrote so much I lost track of myself, so I don't know if this was coherent or not. Anyway, visit
www.homerecording.com and read the stuff there. The site has some very useful newbie info, FAQs and tutorials. They also have a forum, with a special "newbie" section.
Where to Start? 
Well... I don't know more than what I've read on the Internet... I did a whole lot of research before buying my home studio equipment.
