If you just use 1 computer there's no need to use a router, the router is designed for sharing one internet connection or sharing between computers.
Although it's true that adding a router adds an extra security layer, first hiding your computers from the rest of the internet, since it performs NAT (network adress translation), from the outside you can see just 1 connection eventhough at the inside of your network it has many computers connected.
Second: the SPI (stateful packet inspection) firewall which checks evey packet sent and recieved and verifies it was solicited , and the destination is correct, that it's nor corrupted, etc.
So it's an extra layer of security, but if you just have 1 computer and won't share your connection you don't need a router, just get sure to have a good software firewall, and don't just rely on Win XP SP2 firewall, nor Symantec Internet Security firewall, i shall recommend for best Zone Alarm, Black Ice, etc.
Consider that your protection is as good as updated it is, so your software firewall, would be updated every week or more, but the router firewall is not updated so don't just trust on having a router firewall, it gives you an extra layer of protection and make it harder to hack your network, but won't stop virus , spyware and others.
Also you can look for all in one solutions, which includes : router, firewall, antivirus and spyware protection in a box.
PS: also check my other post here