Does Apple provide disk space that I can back my hard drive up to? As I understand, with Time Machine I need to provide a drive to back data on to.

Is there also an option to use a server managed by Apple to back up on to?

Either using Time Machine or another program.
The Apple Time Capsule is a network router and network storage server with a 2TB or 3TB hard drive.

When you said, "server" do you mean something you can own and run in your home, or something that exists at a remote location, managed by a storage company, that you access over the Internet?

Internet storage is not suitable for backing up your entire Mac, because moving that many files of that size through your Internet connection is far too slow and inefficient.

You need a local physical hard drive in your home or business to regularly back up your entire Mac using Time Machine. You can supplement this with an Internet storage service like Mozy or Carbonite, but that should only be used for a critical subset of your documents and data, because of the time required to back up the data, and the high cost.