A turntable, amp/receiver to plug it into, a sound card on your PC, a stereo lead with the proper connectors on it (RCA type at the amp end, and probably a 1/8" stereo plug at the computer end) to go from the amp to the input jack on your soundcard at the back of the computer.

Plus software to "record" the input and digitize it to a .wav file. I like CoolEdit, but I think Windows Sound Recorder (included with Windows) will do the job too. I like a better audio editing program to be able to apply noise reduction, audio editing, a bit of EQing and other filters to the sound data. Make sure you record the .wav file at "CD Quality" audio setting (16 bit, 44.1 KHz) to be able to burn the audio to an audio CD.

You'll also need enough hard disk space to receive the audio data. A CD quality .wav file requires at least 10 Megs of space per minute of recording. You'll also want lots of extra space for editing too (breaking the long recorded side of a vinyl LP into individual tracks, maybe converting them to MP3, etc.) A fast processor and lots of RAM helps too, to handle the approximately 22 minute wav file each side will result in. It has to wait in memory until you save it and the less swap file Windows has to deal with till then, the better.