Get access to a computer. You will need a reasonably fast processor and at least 1 GB of RAM.
Access a digital audio workstation (or DAW). If you own a Mac, then you already have a very decent DAW called Garage Band. Some other very popular DAWs are Logic Express/Pro (Mac only), Sonar, FL Studio (PC only), Cubase, Ableton Live (PC and Mac), Pro Tools (only works with Digidesign or M-Audio interfaces). If you are looking to make electronic music, only then you should also look into Propellerhead Reason.
Get an audio interface. A basic interface usually consists of two input pre-amps that go into two analog to digital converts, two line outputs (left and right) and a headphone output. they are available in USB, firewire, PCI, etc. Usually when you get an interface you will also get a lite version of Cubase, Ableton Live, Sonar, or Pro Tools. This should solve your DAW problem if you are a PC user. Some popular brands are Apogee (Mac only), Digidesign, M-Audio, Tascam, Presonus, Edirol, Yamaha, etc.
Start recording. There are two ways of recording into your computer. One of them is using a microphone (condenser, dynamic, or ribbon) and a pre-amp (usually included in your interface). To record this way, just connect the microphone to the XLR input of your interface, turn on the phantom power (+48V) if you're using a condenser microphone, and set the gain to where it doesn't clip on your DAW (go over 0db). If you want to record using an external pre-amp, make sure you bypass your interface's pre-amp and you have phantom power turned on in either you interface or the pre-amp. The other way to record is by using the direct inputs (also known as instrument inputs in most interfaces). This is usually used for recording guitars, synthesizers, drum machines, or any external sources. To record this way, just hook up your guitar, synthesizer, or whatever you are recording straight into the 1/4 input of your interface and set the gain to where it's not clipping. If you...