Most desktop PCs and many laptops have a Line-In socket for plugging in other audio devices.

Use this to plug in your VCR or record player preamp into the computer. From there, it's simply a matter of capturing the audio, and encoding it.

A program I like to use for this task, is Audacity -- downloadable free-of-charge from http://audacity.sourceforge.net/ for Unix clones, Apple MacOS X and Microsoft Windows.

The interface, as far as recording is concerned, is pretty self explanatory. I generally record my LPs in one take -- hit record, put the needle down... wait for that side to finish, flip the LP over and put the needle down again, then hit stop when the LP finished.

Once recorded, tracks can be split via the "Split" menu item in the "Edit" menu. Simply place the cursor between two tracks, click, then press CTRL+End (not sure what it is on Mac) to highlight to the end of the recording -- then select "Edit->Split" to split the track. Once complete, use the "Export All" item of the "File" menu to save each track separately.

Then use whatever encoding program you like to convert to MP3.