TCP/IP Idiot's Guide Required

OKay, who understands IP? I want a simple explaination as to why I can log into LMR with two different usernames from two different PCs in my house, when they're both attached to the same router. The router assigns them both unique non-routable IP addresses on the LAN side of my network.

If I log into two sessions on LMR using two usernames, the more recent of teh two connections takes over the older one and that session becomes the same as the new one.

Once the traffic passes through my router, it's almost as if my two PCs have two different IPs.

This kept me awake all last night.