Apple - Why is my Mac constantly trying to connect to another server?

Check System Preferences>Accounts>Login Items to see if there is a disk that is supposed to mount when you login. Make sure you check all of the user accounts.

I'm pretty sure this was happening because when I logged out, I was running an app which was located on the other computer.

I logged off and unchecked the "Reopen windows when logging back in" box, which stopped Lion from trying to reopen the app the next time I logged in.

I suspect you are connecting to a legacy file server as described in the kbase article below.

http://support.apple.com/kb/HT4700

We saw similar pop ups but clearing up our mounts addressed the issue.