An authentication system is how you identify yourself to the computer. The goal behind an authentication system is to verify that the user is actually who they say they are.
There are many ways of authenticating a user. Any combination of the following are good examples.
Once the system knows who the user is through authentication, authorization is how the system decides what the user can do.
A good example of this is using group permissions or the difference between a normal user and the superuser on a unix system.
There are other more compicated ACL (Access Control Lists) available to decide what a user can do and how they can do it. Most unix systems don't impliment this very well (if at all.)