The fundamental difference between vinyl and CDs is that LPs use analog audio while CDs are digital. Technically, digital sound is of a higher quality, but some people actually prefer the audio effects of records. Record players work by tracing a needle over precise grooves in the vinyl, producing sound. CDs, in contrast, are covered with countless microscopic "pits," which are then read by a laser.
One way of converting vinyl to CD is manually. The process requires a sound card, turntable, a pair of cables and CD-burning software. Most PCs already come with CD-writing software installed. You need an RCA-to-mini headphone jack cable to link the sound card and amplifier as well as a normal RCA cable to connect the amplifier to the turntable. The audio from the vinyl is processed through the sound card and burned on CD as a WAV file.
GoldWave is a free, downloadable audio program that records records on CD. You'll still need to run a cable from the turntable to the PC. GoldWave will record the entire record as one track, which you can then break up into separate tracks before finalizing the CD. You can also adjust audio levels as the record is burning (GoldWave displays meters that read audio input) and add effects like hiss reduction.
A more low-tech way is recording sounds straight from the vinyl into an external microphone hooked up to your computer. This will produce much lower sound quality but is a more streamlined process. Simply position your mic near the amplifier, set it to record, play the record and save the file. Then use standard CD-burning software to create a new CD.