Answer: Any transmission (streaming) of copyrighted music is going to fall under at least one royalty classification. These categories - overseen in the U.S. by a trade group known as SoundExchange - include commercial webcasters, educational, non-commercial, etc. For instance: a small, non-commercial webcaster/microcaster could pay as little as $500 per year.
This is assuming you are providing your own Internet streams through your own server(s). But, if you are streaming an Internet radio station through a third party like Live365.com (Direct Link), you pay a monthly fee to that company and they pay the required royalty fees to organizations like SoundExchange, BMI, ASCAP, and SESAC.
Live365.com (Direct Link) allows you to start a Personal Station for as low as $5.95/month which will support 3 listeners at once. On the other end, a $99.95 package will support up to 100 listeners.
Downloaded Music as Content: When you download copyrighted music from a service like Rhapsody, you are paying for the right to listen to and enjoy the music for your own personal use. Your usage rights do not extend to using the music for any purpose other than your personal usage, such as streaming it on an Internet radio station.
Music You Can Use: There are many locations on the Internet which allow you to download music that can be used without paying royalties. That's because the music is from artists or bands who want the exposure and are willing to trade that for a royalty payment. Blogtalkradio.com has a list of places to look.