Macs often have problems with network routers. You might want to buy an Apple certified router. I'm not an expert on getting Macs to work with routers that aren't confirmed to work with Macs, so I don't know if your router is supposed to work with them fine and I don't claim to know for sure if it should be replaced or not, but it might be necessary.
Also 192.168.1.1 is simply the router's IP address on your network. That's the usual IP address for devices that share internet to other machines on their network if I remember correctly. It doesn't refer to anything special about the router.
Are you sure that address is correct? There is nothing special about Macs, they should (and do in my experience) work with any router. And the configuration interface is just a web server, so again nothing special.
Just for clarification, how do you know that it'address is 192.168.1.1? Check the default gateway in the network settings; that will be the router's address.
Agree w/ Ijack, Macs don't have problems with network routers unless there's faulty hardware involved.
It sounds like you're trying to find your router using a default IP address, but the IP address may have been changed. Also confirm that your firewall isn't blocking access out; this varies between different versions of OSX.
I've had several problems with my routers with Every Mac and Macbook that I've worked with. WiFi would often lose signal but be fine on the Windows computers that were being used at the same time (even right next to the Macbooks) and I've had other issues. Some routers were more forgiving, but I've had several that simply didn't work well, if at all, with Apple's computers.
A fully functional Mac will never have an issue connecting to a standard network. Ijack is spot on. The OP needs to verify the IP.
However, a search of available networks, and the use of DHCP would negate even needing to know the router's IP address to connect to it. This assumes that the OP has the proper passphrase to connect to the network (if it uses one).