Your Mac should default to using an automatic address through DHCP, but that begs the question "what's supplying the DHCP address?" It would be smart to let the router do that, so I have to ask how your router is set up, and, of course, which model you have.
Another angle is whether or not you have any security features running. When getting things started, it's often good to turn off all security features until you get the basic connection working, then add security features one at a time. I usually start with a wide open access point until I get a reliable connection, then add encryption at both ends (which usually takes a little fiddling to get going), and then finally enable and configure MAC address filtering.
Finally, how is your browser set up? Is it looking for the right information, such as DNS server and so on? Is your AirPort card getting anything from the router? Just because you have a good signal doesn't mean that the computer is actually talking to the access point.