Here it is in simple terms:
Gateway mode: NAT enabled on the WAN port
Router mode: NAT is disabled on the WAN port. The WAN port is still functional.
The key to getting router mode to work is that there needs to be a static route added to the upper level routers to route the traffic back to the subnet behind the router that is in router mode. One will notice that if just the router mode is enabled that you will be able to ping all the interfaces on the router but nothing past that. One will also be able to ping anything using the diagnostic abilities of the router itself, but that nothing behind the router will be able to communicate with anything past the router. This is because the upper level routers don't have a route to the subnet behind the router mode router (unless dynamic routing protocols are used).
As far as I know this is the case with all Linksys/Cisco home and small business routers, or any router with a gateway mode or a router mode. I've only tested this on a WRT router and a RVS4000 small business router.