The simplest way may not be the hi fi way but,
1. turntable connects to phono preamp input--The phono preamp contains RIAA equalization curve, without the curve of equalization your records will sound wrong.
2. phono preamp output connects to adapter. The adapter changes RCA plugs to 3.5 MM stereo plug male.
3. Adapter connects to the line input of the computer. Make sure the input you are using (on the computer) is set to LINE level, not microphone level. You can change this level with the audio software, provided by your motherboard or computer manufacturer. (your audio software is probably already loaded)
4. Install an analog to digital converter application, such as Sony Sound Forge.
Once you have played and recorded the LP, you can save the file as: MP3, or other format.
The recording may be played with windows media player, etc...or you can save it as a regular CD audio format and play it on a conventional CD player.
TIPS: Your phono stylus and tracking weight are important factors in sound quality. A crappy stylus = a crappy sound transfer. Incorrect tracking weight = no bass frequencies.
Of course you can use another method besides RCA to stereo plug, you could buy more expensive equipment such as USB turntable, etc...
you could use free software, but I don't recommend it. Sound Forge is a studio quality, bit rate converter, etc...which will come in handy for many audio projects.
DANGER: do not attempt to connect the speaker outputs of an amplifier to the computer line audio input. It melts too easy.
I'd tend to agree with the above though I would mention that I use a conventional stereo amplifier which has a phono input and feed the output from the Tape Out socket of the amplifier to the audio in socket of the computer using the sort of RCA/Phono to 3.5 minijack normally supplied with soundcards.
I've used the free program Audacity which will convert analog to digital WAVs which can then be burned as audio CD -- or Audacity can save the files as MP3s to use on a personal stereo.
More recently I've been using Sony and Philips CD Recorders which are linked via a hifi amplifier to a Technics SL1210 turntable. This is a nearly ideal system for the highest fidelity as the DACs in the CD recorder are higher quality than those in a sound card (and there 's none of the electronic noise you get in a computer). The Technics is the standard disco DJ turntable notable for fast start and stop which is useful in this application.
The only problem is feeding the CD recorders which use so-called Music-type CDR blanks which cost much more than computer CDRs and are harder to obtain. The way round that is to buy Music CDRWs and re-use them having transferred the content to computer disks (which will play on a CD player).
The answers above are quite correct but the recording program does not do the analog to digital conversion. This is done by your soundcard.
A program to transfer vinyl to PC that is very good is called Vinyl Studio. It does post processing to remove noises. There are also some recording programs that allow you to connect the turntable directly to the PC and applies the eq digitally.