Avatar login server
Avatar Name Server: is the overall grouping of grids.

Select the group of grids that use the same login server. (This allows you to use the same username in all the grids that support using the same login server, in addition to a local Opensim, if you have it set up to allow any password)

For example the SecondLife login server will accept your Second Life avatar name and password, and this will allow you to log onto the main grid and the beta grid. Also if you have a local OpenSim with no password (or the password set the same as in Second Life), you can log into there too with the same avatar.

Note: For ease of use, I have already set up a local OpenSim grid server in each of the default Avatar Server groups. What this means is that if you have a default OpenSim (with no password required), you can log into it with any of your avatars from any grid (of course it is name only, but it is nice to have your own name on your own sim).