Edit: I was bored and this post got rather long. If you already know any of the info to follow, I wasn't trying to be condescending - I just went with the assumption that you were starting from scratch, as so many do, so feel free to ignore/correct anything you have an understanding of
What type of music are you looking to record, and how do you want to go about doing it?
If you want to record piano, at the very least, you would either need mics and access to an acoustic piano (and a portable computer if you plan on recording on-location), or a midi controller and midi interface to the computer. Not sure what your knowledge of all that is, so we won't get into details just quite yet.
Beyond the piano, if you want to record vocals, you'll need a mic of some sort, and an interface to the computer that accepts mic inputs. You can buy interfaces that handle both midi input and mic inputs, like the following:
http://www.musiciansfriend.com/product/Digidesign-Digi-002R-Music-Production-System?sku=704168
Basically, the interface takes any inputs you have (microphones, midi instruments, guitars/etc that are plugged straight into the machine instead of amplified), and converts that signal to digital to record and edit in the software of your choosing. They can connect to the computer via USB or Firewire, depending on the company's choosing. Most these days are firewire, as far as I can tell. This is en expensive example - you can find much cheaper alternatives, but it gives me all the components I think need to be explained right out of the gate.
First, you have your midi interface - the midi controller (a keyboard), plugs in to the midi in/out ports, and instead of sending a musical signal through the wire, it sends information into the machine regarding what keys you hit, how hard (velocity), what pedals you ahev pressed down, and some other information. Then, whatever software you are using interprets this data and plays back the sounds associated with each key in the software you use (Reason has a nice package of synths you can set to respond to your midi input, so you have a whole range of pianos, key synths, etc).
As for mics and the interfaces, you have a couple things at work. First, the microphone. I'd go over to the Mics part of the forum and read the sticky, to begin with. You have a few basic options: dynamic, condenser, ribbon. I won't go into the differences as I cannot do the topic justice in a brief paragraph, so I'll just explain the connections to the computer. The mic gets plugged into a mic 'preamp' - basically an amplifer designed to take the extremely low-output signal from a microphone, and push it up to the level that a recorder can store and reproduce.
The unit listed above has preamps built into the microhpone inputs, but you can also buy external mic preamps, since they are a big part of the sound of the recording (or lack of added sound, as many try to achieve), and they can vary from 10's of dollars to thousands (and higher?). The mic pre at the very least then runs into whatever recorder you are using.
At this stage, you are either looking at analog or digital. Analog recording usually uses some sort of tape to record the signal you have produced, and stores it magnetically. With digital, the signal gets run into a converter that takes the analog signal and changes it to digital for storage on the computer. Any time you record into the mic input on your computer, you are performing an analog-digital conversion (not a very good one mind you). When you listen back to what is on your PC (or Mac), you are converting the digital signal back to analog for the speakers to reproduce.
The difference between digital and analog? I'm not going to get started on that debate - regardless, digital will (presumably) reproduce the exact same sound you recorded every time, as a digital copy does not get altered when it is accessed from your computer's memory, whereas playing back audio on magnetic tape loses quality over time, since the tape is rubbing over read/record heads, which wears it down. There are supposed advantages to analog... many say it sounds "better" - I purport that it is simply different, but again... search around for the digital-old "analog vs digital" debate.
Once you have your setup (mics, interface to the computer, the compy itself, and any other toys, like midi controllers, etc), you have to start learning mic usage and placement, how to set gain (the volume on the preamp, the volume settings in software, etc), etc. There is a wealth of information here on all these topics.
Software? I don't use any multitrack software since I record to digital tape machines and mix on a clunky analog mixing console, but ProTools is one of the better-known pieces of software for recording multiple tracks and mixing them together. Check out the computer recording area for plenty of threads along the lines of "Best software for recording" / "Favorite DAW software", etc etc.
Any of these topics have been drastically skimmed over, so start searching for the various topics, and read as much as you can here - its honestly not a HUGE learning curve to get to a semi-educated point, with a combination of asking questions, experimenting, and reading as much as possible. Good luck - I hope my rant wasn't too boring or inaccurate (and if it was, somebody correct me!)
