Client mode wireless is generally used to "retrieve" an Internet connection (using the Wireless portion of the router) from another router, and then share it out to the LAN (wired switch ports).If you have a wireless router connected to the Internet in one location and would like to connect wired clients in a remote location to allow shared Internet, one solution is to use Client Mode Wireless. This will not require the host router to be running DD-WRT firmware. The WRT running DD-WRT firmware and configured to run in Client Mode will connect to the host router as though it were any other wireless client, and will share the wireless connection out the the LAN (wired switch ports) as though the host router were connected directly to its WAN port. This means that the host router and the Client Mode router will be on seperate subnets. NAT will be used between the routers, so when port forwarding is needed, it will need to be configured at both routers, not just the host router. Devices connected to the Client Mode router will not be able to use the DHCP server from the host router. Also, a device connected to the Client Mode router should use the Client Mode router as it's gateway and DNS server.
A router in Client Mode Wireless will not be visible as an access point. It will not accept any wireless connections from client devices. If you'd like to wirelessly "daisy-chain" routers to extend your range, you'll need to use repeater mode