You want to record a performance? Maybe you want to make a "demo" recording of yourself or a group?  With a limited amount of equipment you can make a very good recording. Let's first discuss what you are going to record and then some equipment options.

"Recording Session"

This is a recording which is in a controlled environment. There is no audience, so you don't have to worry about extraneous noises from listeners. You have selected the location for the recording..Maybe you can't get the location you feel would be best, but at least you were able to select the location. As a result, you can set the musicians up specifically for the recording, taking advantage of the acoustics without worrying about how they will look to an audience. You might want to separate them more than normal, or face them in certain directions. The main point is that you have much more control over the environment, both the physical setup and, most important, you can stop and make changes or re-record something that has problems.

"Performance"

The temptation is to say "live performance", but if it's a performance I would hope it's "live". Here you are faced with an inflexible setup, one that was done with an audience in mind. So you must work around the limitations. You also have to be concerned with noise from the audience (and you can be absolutely sure there will be noise--just hope the crying babies have been left home!).

There are other conditions to consider about what you are recording.

Acoustic Performance vs. Amplified Performance (or Recording Session)

With an Acoustic Performance there will be either absolutely no electronic amplification or, at worst, one or two instruments will have their own amplification. You can normally expect this kind of performance for classical music.
If you are recording a performance of non-classical music, you will more likely encounter electronic amplification. While folk music, jazz, and even some country music come in variations that include non-amplified performances, when a performance is involved the amplification of part of the group (especially vocalists) may be needed. Rock (and most country music) essentially requires amplification.

Acoustic performances can mean better sound, but require more attention to the acoustic space where the performance is taking place. If there is an audience, the audience noise will be heard more easily. Amplified performances may make an audience (if there is one) less easy to hear, but, on the other hand, the audience may make more noise.

Certain factors about the recording equipment are very important.

Recording digitally (using DAT or a hard drive recorder such as the Alesis ML9600) means you will be able to make a recording with less noise and a wider dynamic range. Once you set your recording level (to handle the loudest peaks) you shouldn't change it. (Of course, if you are recording a performance you will have to make your best guess and make slow corrections during the performance if you are too far off.)

Probably the most important part of the recording chain is the microphones. You only need two for a great recording, but how well the mikes "listen" makes all the difference in the world.

Good results are not difficult to obtain, as long as you are pay attention to these basics and remember to listen.