What is the difference between an Authorization ID and a Authentication ID?

Authentication is the act of proving who you are. “Hello, I’m Dave. To prove it, here’s my password: Foo.”

Authorization is the act of deciding whether to grant access to resources.

"I'd like to read Kellie's mail for her."

In the example, I’m trying to read my wife’s mail. I supply my own username as the “authentication identifier”, my own password (Or biometric scan, or whatever else is required to prove I’m really me, with whichever mechanism is in use), and my wife’s username as the “authorization identifier”.

At no point need I know my wife’s password - instead, either Kellie or an administrator needs to explicitly state that I am allowed in “as Kellie”. Once I’ve logged in, all the access checks are done against Kellie, not against Dave, because I’m acting for her. To all intents and purposes, after the authentication exchange itself, the server can simply forget about who authenticated - it’s not important any more - and concentrate on who needs to be authorized.

Another, more common example of the use of differing authentication identifiers and authorization identifiers is in the design of many proxy systems. You authenticate perfectly normally to the proxy, authorizing as yourself. The proxy then authenticates to the master as itself, but supplies you as the authorization identifier, thus getting all the right access checks done at source, but not having to have access to your authentication credentials. Finally, some mechanisms don’t support passing a distinct authorization identifier, and for most its optional, and defaults to the case that most people are familiar with, where authorization and authentication identifiers are the same.