Digital music consists of a large number of ones and zeroes. You can create digital music on a PC or with various digital musical instruments, but most digital music starts out as analog music. When you go to Alice Tully Hall in New York to hear a string quartet you are listening to analog music. If you want to listen to it later at home you need to have a way to capture it, store it, and replay it. In the "old" days this was done with vinyl records and later with acetate tape. Today it is mostly done with CDs (compact discs) but increasingly music will be stored in the form of digital files such as MP3 (MP3 is one of dozens of different formats for storing digital music).
Analog music is captured with microphones and recording equipment. It is frequently stored on tape initially but usually ends up on a CD in CD-DA or digital audio format. This is done by electronically sampling the sound 44.1 thousand times per second and capturing 16 bits (ones and zeroes) of information about the characteristics of that second. The result is 88,200 bytes (a byte is 8 bits) of data for a second of music. Multiply this times two for stereo and you have 176,400 bytes of data per second. Multiply that times 60 and you get 10.584 megabytes (millions of bytes) per minute of music. A CD holds about 660 megabytes of data so that gives you approximately 62 minutes of music on a CD.
OK, so what is MP3? There is a group of experts (from IBM and other companies) called the "moving pictures experts group" which created a standard called MPEG. MPEG has various "layers" which specify how audio or video can be compressed. Compression removes bits from the sampling process that are not essential or even recognized by the human ear. A brief pause in a song, for example, can be eliminated or compressed and then decompressed later when it is played. The result of compression is that a much smaller amount of data needs to be stored. MPEG layer 3 describes a particular standard for achieving high quality sound with compression. It results in a compression ratio of roughly eleven. In other words with MP3 you can store roughly 11 hours of music on a CD. It also means that CD music can be stored on a PC in about one eleventh of the space required if it were not compressed. It further means that now that it is compressed it has become practical to send it over the Internet in a reasonable amount of time. The result is that file sharing programs such as Napster, which was designed to share PC files, instantly became a very convenient way to share MP3 music files. BitTorrent is taking this to the next level.