If your turntable has a pre-amp it can be connected directly to your PC sound card. Your turntable will have PHONO outputs like this:

So to connect this to your PC sound card you will need an adapter. The adapter converts the two phono plugs to a 3.5mm jack (headphone plug) and looks like this:

Plug the two phono leads from your turntable into the adapter and then the adapter into your sound card MIC or LINE IN port (LINE IN is ideal).
As below:

Connect your turntable to the mains and turn it on (if it has a power switch). You are now ready to record so you can now skip ahead to the recording part.

Turntable without Pre-Amp option or connected through Stereo

If your stereo has a REC OUT (would have originally gone to cassette recorder) connection like below you can run a lead from this to your PC.

You can use a normal double ended PHONO lead with the adapter shown in the "with preamp" section above or you can buy a lead which goes from PHONO to 3.5mm jack shown below:
or use normal PHONO lead and adapter:

If your stereo doesnt have a REC OUT connection you can connect from the headphone socket. This will just need a double ended jack lead. Stereo headphone sockets are normally the larger 6.5mm but a small adapter works great. You can see the double ended lead and the 6.5mm adapter below:

So connect one end to the headphone socket and the other to your PC sound card LINE IN.

Connecting via the headphone socket isnt ideal and you will need the set the volume quite low so you minimise distorting. You are now ready to record.