Beats 1 Radio
One of the biggest parts of Apple Music is Beats 1 radio. This is not part of the Beats Music app, which is now dead. Beats 1 is a global live radio station with real DJs choosing what you hear. It does include popular tracks, but there is a much wider range of music than you get on a local Clear Channel top 40 station that plays the same 25 songs on a loop.
You can see the Beats 1 schedule on the Apple Music tumblr, where you'll be able to figure out when your favorite DJs are online. Although Apple promises 24 hour radio, the service is already seeing repeats form the first 12-hours of content.
If you want to hear a song on Beats 1 radio, you can call in a request using the numbers listed on the Apple Music tumblr. Of course, you can also search Apple Music for the song and listen immediately, but that may take some of the fun out of a request.