In the moderation section, it says some of my members have multinicks.  Is this based on IPs, or does it actually know if it's a different computer?  When I check on one person, it lists the multinick three times to the same person. So, both of them accessed 3 times from the same computer?  Or they both access from 3 different computers?  Or are they just on the same network?

From what I know, the Multinick control is not based on IPs. Indeed, the IP address usually changes at each new connection.
Regarding the multiple info you see about multinick it may be linked to connection on different workstations.