Recording is a whole different ball of wax from performing. Much of the basic equipment is the same, but the additional recording equipment can get complicated (and expensive) enough to make your head spin.
Start by figuring out just how sophisticated a recording you want to make. If you only have a few parts to record, and aren't overly concerned about sound quality, a basic 4-track recorder that records on standard cassette tapes will be fine. If you want to produce something that will be distributed on CD, digital 8-track or hard disk recording is your best bet (although there are those who would argue that tape is better. More on that later.). What kind of recording you want to make will also determine how much money you spend on effects and amplification, microphones, as well as the sophistication of your mixing board.
But first things first.
There are a few basic components you'll need to record electric or acoustic instruments:
The first order of business is to have something to record on. The cheapest route is a 4-track recorder that uses standard audio cassettes (but plays them at double the normal speed to increase fidelity). You can find these in a variety of configurations, from the very basic with only a few inputs and controls to direct the mixing of sound, to very sophisticated ones with submix busses and lots of inputs. This is a middle-of-the-road model that I've had reasonably good luck with (although it has a few critical drawbacks, the largest of which is that the auxiliary busses can't be monitored in stereo.), the Fostex 280:
The main features you see on this model are 8 inputs (rather than the four you get on cheaper models, or the 12 you get on more expensive ones), 2 auxiliary busses (so you can send the signal from individual tracks to offboard effects units, like reverb), memory locators, and so forth. Sound quality is fine for basic demos, but would take much tweaking to sound good reproduced on standard-speed cassette, and definitely has too much noise for CD.
If you want sound quality that can actually be mastered to CD with professional-sounding results, go digital. 4-track cassette decks are "analog," meaning they store information on the tape as direct reflection of the original sound (in this case sound turns into a magnetic wave that's imprinted on the tape). But digital tape recorders convert the sound into numbers -- 1's and 0's that can be stored on large-format tapes without the additional noise that's inherent in standard analog recording. And more importantly, digital recorders are getting dirt cheap (well, relative to what they used to be).
This is an Alesis ADAT XT, a $2500 recorder that's probably the most popular recorder on the market today. This recorder uses SVHS tapes, just like those used in SVHS VCRs. Many professional studios use these recorders, but even cooler is that many small home/project studios and individuals can now afford to use them. So it's getting that much easier for individual musicians to produce and record their work without the expense of a full studio.
There is some debate, however, over the absolute quality of analog vs. digital recordings. Purists argue that digital strips the sound of much of its warmth due to the sampling process (in converting the sound to 1's and 0's, a digital recorder looses some of the original signal because it can only "hear," or sample, a certain percentage of all the sound produced.). Others argue that the vast majority of people won't ever hear the difference. In either case, the only real issue at the project studio level is how expensive an analog tape deck you can afford. Most analog decks that rival digital recorders in noiselessness (and have similar features) are a good deal more expensive. There is also no way to achieve on analog tape some of the editing effects that are possible with digital recorders. My opinion is that until you're in a high-end studio, digital is the way to go. And in the case of acoustic recordings, where warmth is a large concern, remember that a really good microphone will add more to the overall quality of the recording than a digital recorder will subract from it. In any case, be aware that analog recording is still very much alive, and you may want to spend some time listening to the differences.
Once you have something to record on, you'll probably need a mixer. Many 4-track recorders have small mixers built in, but the ADAT does not. (Another drawback to most 4-tracks is the fact that their mixers are quite noisy, and put more hum into the signal than stand-alone mixers). You'll also find that most stand-alone mixers have many more features than 4-track mixers, and allow you more flexibility when mixing the various tracks of your recording. This is a Mackie 1604, a 16-channel mixer that is very popular for both its price and performance (it's suprisingly quiet, and has many features you wouldn't expect for such a small mixer).
If you're recording any acoustic instruments, you'll need a good microphone. In most cases, you'll get the best sound for acoustic violins and like instruments with a condenser mic. "Condenser" is the old word for capacitor, and refers to the fact that this kind of mic uses that electrical component to actively respond to the sound (thereby being more responsive), rather than passively responding to it in the fashion that a dynamic mic does. Condenser mics also typically have larger diaphragms than dynamic mics, and can accordingly respond more sensitively to sound. The mic shown is a Neumann large-diaphragm condenser.
At some point you'll want to actually hear what you're recording. You can use a good set of headphones (I recommend the AKG 240 monitors. About $125 and well worth it.) as you're recording individual tracks, but to mix you'll need monitor speakers. Large studios will have several sets of speakers to allow for previewing the mix in a variety of configurations. At home, you'll be perfectly happy with the Alesis Monitor One speakers shown here. But keep in mind that when buying speakers, you have to get something to drive them, such as....
Gotta drive your monitor speakers with a good amplifier. About $300 will get you the Alesis RA100 shown here. My only real recommendation with regard to monitor amplifiers is that you get one that's quiet. Some amps are designed to drive speakers in a club or bar, and as such have a fairly loud transformer hum. If you're recording at home, you'll want an amp that doesn't do that, or that you can't hear because it's off in a rack someplace away from your listening area.