Hello Scott,
Actually, you are asking about port roles, not states (sorry for the nitpicking but these two are strictly differentiated in RSTP).
An Alternate port provides a backup of your own Root port. If your Root port fails, the Alternate port is allowed to immediately transition into the Forwarding state and become the new Root port (in essence, the Alternate port is the one that receives the second best BPDU).
A Backup port is a backup of your Designated port into a particular network. You won't see the Backup port role often because it would require connecting your switch with multiple links to a shared medium, say, a hub, which is not done commonly anymore today. If your Designated port into a shared segment fails, the Backup port will take over the role of the new Designated port. However, a Backup port does not immediately transition to a Forwarding state if the Designated port fails. The Backup role is merely providing a defined role for this kind of port (so that each port can have some role assigned) but for shared segments, there is no rapid convergence. Rapid convergence in RSTP can be achieved only on point-to-point links.
I suggest also reading the following document:
http://www.cisco.com/en/US/tech/tk389/tk621/technologies_white_paper09186a0080094cfa.shtml
Best regards,
Peter
