MP3 (technically, MPEG Audio Layer-III) compresses a digital audio file to about 1/10 of its original size (it can vary depending on the bit rate), while retaining near-CD quality, so that download times and required storage space on a computer are much more manageable than before. For example, you can download a 4-minute song in less than 15 minutes, as opposed to more than 3 hours in CD audio format (33.6 kbps modem), and a CD holds more than 12 hours of music in MP3 format, as opposed to the usual 74 minutes. You can download to your computer from the Internet, convert to MP3 format from prerecorded music in other formats, or record from scratch. Then you can play music right off your computer or on an MP3-friendly player.
Streaming audio, on the other hand, transmits music, pieces at a time, to a temporary storage place (buffer) in your computer. After a short wait, the buffer is filled and music "streams" out through your speakers. The quality can be iffy, depending, among other things, on the speed of the particular Internet connection.
You can, of course, buy music online in the form of traditional tangible media, such as CDs and tapes, but now you can also download (or upload) music files (in formats such as MP3) or listen to them as samples or on Internet radio (in the form of streaming audio). Sometimes it’s free; sometimes there’s a small fee. Sometimes permission has been given; sometimes it’s outright piracy. Try to use reliable sites by reading the fine print, checking with ASCAP, BMI, or SESAC, and generally staying informed on the latest news. (Check in with Wired’s MP3 Rocks The Web.)
The new media has led to a more custom environment where, for example, people can listen to song samples before buying or downloading, and they can determine their own playlists. It is no longer necessary to buy entire albums, when you can choose one or two favorite songs to download to file or physical media.