A method of compressing music (mainly from CDs) hugely (usually by about a factor of ten), without reducing the sound quality noticeably at all. Generally, before compression, the data in WAV format (the commonest Windows audio file format) is over 10 megabytes a minute — afterwards it's about 1Mb a minute (that's at 128kbps compression.) You can fit over ten albums on a normal computer CD-ROM.
Any drawbacks?
For record companies and artists, yes. MP3 technology makes it incredibly easy to pirate music — plus if you have a CD writer, you can get many hours worth of music onto one CD... of course, it won't work on a normal CD player, but that's not much of a drawback if you have nice computer speakers. If you're responsible, you'll only MP3 copyright music if you own a copy of it — that way you're still in a legal grey area, not actively breaking the law.
Plus-points?
MP3 makes it really easy for young bands to get exposure — they might not get any money from releasing songs in MP3 format, but they're more likely to get heard. Better to release a taster of a few tracks as MP3s, and then give order details for a full CD on a website.