a CD burner is a CDROM drive that can write CDs. You can replace your original CDrom drive with one or get one that is in an external case and connects via USB, SCSI, or Firewire (external ports put simply). With a Cd burner, you can put files on a CD. You can also copy a music cd so that if plays in a normal CD player.
MP3's are CD music tracks which have been compressed (made smaller to take less room) using a special algorithm. For example, CD audio takes 10MB of space per minute of audio. A MP3 is much more efficient, using approximately 1MB for 1 minute of audio.
An MP3 player is a small, compact device which reads the compressed audio file and converts it to an analog signal for your headphones.
MP3's can be made using MP3 conversion programs which let you choose audio CD tracks and convert them to MP3's which you can store on your computer's hard drive. MP3's are good because they are digital, meaning that they are at the same quality as a CD (if you choose that quality from the various qualities available).
A Cd burner should NOT be more than $140 for a 10x4x32x (10x write, 4x re-write, 32x read CD). External burners are more expensive as they have a case, circuit board, power supply, and interface cable. They are generally easier to install because you don't need to open your computer; just install the drivers, plug in the drive and go.
Hope that clears things up for ya!