You have no control over your external IP address--the one that the modem gives whatever is connected to itself (in this case the AEBS's external network connection). To computers outside your network, everything within appears to have this IP address.
The internal IP address of everything on your internal network, however, is set (usually) by the router; the whole point of port forwarding is so that when a computer outside your network needs to initiate a connection directly to a computer within, the router knows which specific internal computer to forward the connection to.
To do this, you create a port forward; you tell the router "whenever someone tries to initiate a connection to port XXXX from the outside, route that connection to port YYYY on IP address ###.###.###.### on the internal network." Obviously for this to function the way you want, you need to have a consistent IP address on the internal computer.
Since most home routers are set up to use DHCP (such that the router gives an available IP address to any connected computer that asks for one), the easiest way to do this is to go to the router's DHCP configuration section and tell it to always give one specific computer one specific IP address (you specify the computer using its MAC address, aka "hardware address", which is unique for every computer--your router can usually tell you what it is, or you can get it from the System Profiler). That way, the router will reserve that specific IP address for that specific computer, and give it to that computer whenever it connects.
In your case, when you replace your computer, all you need to do is change the MAC address in the DHCP section to match the new computer; the new one will then get the correct IP address and the port forward will work.
Again, with Apple hardware (maybe some other routers as well), you can also use UPnP if the software supports it; that essentially sends a signal to the router to automatically configure whatever port the software requests to forward to that specific computer, and it can be changed on the fly.
This stuff is kind of confusing, but it's not horribly complex to set up once you understand what you're trying to do.