Well, you can do both... it's just not easy. Apache is the best way to go as a server; for a domain, once you buy it you can point it to your IP. However, this means if you're not on a static ip, you'll have trouble keeping it up to date.
Also note that on a home connection your domain is gonna be SLOW. You don't have a huge amount of upload bandwidth.
Totally agree with boookworm99. It can be done, but it will not be fast. If you want to run small web apps for personal use, it can be useful. However, if you want to have a lot of visitors at the same time, it won't help you at all.