Look into getting a no-ip.com, dyn.com, or dnsdynamic.com account. All of these are free dynamic dns providers. So, basically:
1) Set up your server with whatever services you desire
2) Open an account with one of those sites
3) Read the instructions on how to set everything with the relative site
4) Confirm with your ISP that your IP is accessible from the internet/is not behind a firewall
5) Ideally, should be good to go
There will most likely be a lot more steps you will have to be involved with, but these 5 are a general outline of what you have to do for what you want to accomplish. Hosting from your computer is a rewarding experience, but I really wouldn't recommend it if you don't know what you're doing as far as security, uptime, configuration management, and user support go. That's not to mention purchasing/renewal fee's if you want a FQDN.