Charlotte thinks this might be because we only have one IP address.
Sounds likely.
Is this an A&A supplied router? They usually configure them properly out of the box.
How many IP addresses did you ask for? If you've only got the one, then ask the nice guys at support to allocate you some more. If you're not sure, log in to clueless and check the page for your line.
Or stick with the single IP address and configure your router to do NAT, which is evil, but is how 99% of home users connect to the internet (and therefore likely to be assumed to be the only way to do such things when you google or ask on forums). 192.168.whatever addresses aren't proper internet-routable addresses, and require your router to be doing NAT.
To make things even more confusing, there's the allocation issue, which is orthogonal to having correctly configured routing. Usually this is done by the router running a DHCP server, which hands out addresses to devices on request, ensuring that there isn't any duplication (two devices with the same address on a network breaks things). That may be not running, misconfigured or even buggy. Though if not running, I'd expect your devices to throw some sort of (possibly vague) error when you try to connect.
I appreciate this isn't very helpful. If I had access to your kit I expect I could work out the problem and possibly resolve it in about 10 minutes. If you had some more specific symptoms it might be a bit easier to hazard a useful guess, but it's going to take all evening to teach you TCP/IP 101 to a level of actually *understanding* what is (and more importantly, isn't) going on.
Total stab-in-the-dark suggestion that could explain the behaviour as described: Is the router in 'bridge mode'. Make it not be.
Pls to be posting the output of "ipconfig /all" (windows) or "/sbin/ifconfig" (*nix) from (preferably both) computers when in a working, connected, state. That should give us something to work from.