"Music on the computer" is a bit vague, as there are several types of music that can be played on a computer.
The most popular are MP3/AAC files, which can be downloaded for 99¢ from the iTunes Music Store (within iTunes), or can be "created", free of charge, from CDs you already own by putting the CD in the drive, then clicking "Import" in iTunes. This kind of music is stored in files on your hard drive, so the more music you buy or "rip" (ripping is getting music off of CDs as described above), the more hard drive space you use up.
There is also "streaming music," available within iTunes from the "Radio" playlist on the left. This music, unlike the music described above, does not reside anywhere on your hard drive -- it is "streamed" to your computer, which then plays it in real-time, kind of like a radio station. Simply double-click a "stream" in the list to start listening -- no files are created on your hard drive, and when you quit iTunes or press the "Stop" button, the music stops and you're all done.
Stick with iTunes and ditch RealPlayer, unless you encounter a streaming radio station on the internet that you really enjoy that only supports RealPlayer.