In access point mode, it's strictly pass-through. It's like an ethernet hub, only with wireless connections instead of ethernet cables. Any sort of management such as handing out IP addresses has to be handled elsewhere.
In router mode, it presents a single address to the network it's connected to so it looks like a single device. On the wireless side, it hands out IP addresses to the computers that connect to it and translates requests from these addresses to its single network address. This is what you normally use -- the router appears like a single computer to your internet provider.
In client mode, it's acting as a bridge. This allows you to set up a local wired network from it that's joined to another network wirelessly.