The repository database is the one that holds the cube structures and
metadata. This database is used by Analysis Manager and the DSO library.
It really only comes into play when building and processing cubes and
dimensions.

As someone pointed out in a post recently, you can query cubes without
having this database at all.

There are two different connection strings because when you are on the
server you can access this database using "localhost" or a reference to
"C:\...", depending on whether you are using Access or SQL to hold your
repository. When you access AS remotely using Analysis Manager you need
to connect to the server by its network name, or to the access database
via a network share (which is setup when you install and Analysis
Manager knows about)