The router will have two interfaces. One is External (web facing) and the other is Internal (Local Network). The External interface is the one that connects to your service provider. This obtains an IP from your ISP and acts like a regular computer with a firewall (basically).
The internal interface is the LAN facing and it has a private IP. (192.168.1.1 usually) This is the IP your computers will use as a Gateway to connect to the internet. Also, most routers will have a DHCP server and DNS server/relay build in so it will automatically assign and manage the addresses for you LAN. It will also tell your computer what the Gateway and DNS addresses are so your shouldn't have to mess with it. The DNS server/relay in the router will accept domain look-ups and translate those to IP addresses as well. (It will change www.google.com into 208.67.217.230 so your computer knows where your data is going.)
You're not manually setting the IP on the Playstation/PCs are you? DHCP on the router should handle all this without you having to mess with the router or manual IP settings. Using the automatic configuration on the PS3 should connect you with no problems. The only exception to this is if someone set up your router security to only accept connections from specific MAC addresses (a network card hardware ID) or to block any extra connections. If you just pulled it out of the box and plugged it in, it's not likely that any of these are enabled. If it's a wireless router, it also means that you may be sharing your internet with your neighbors unknowingly...
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