no you cannot host your own domain, you need to regester it with a hosting company which can then redirect the domain to another domain, or an ip address. you want the latter.
for sub domains you will need to have a dns (domain name server) running on your network, probably the web serving computer.
if you are happy to just have www.ydomain.com link to your web server then there is no need for a local dns, just run the web server as you are and tell the hosting company to link to your ip address.
if you want sub domains say so and can someone else here please help !hi! with setting that up because i don't have a clue
AMD athlon 64 3000+, 1GB ddr400, 400GB total hdd, ati radeon x700pro 256mb (pci-e) 17" tft(@1280x1024).