I'm not sure what you're asking for, so I'll try to cover about everything.
For a web server you get a server with public ip address (like 188.8.131.52) that runs a web server (Apache for example.) From this moment on, you can access that website by surfing to http://184.108.40.206
Then, you can buy a domain name (mydomain.com) and have it point to that ip address 220.127.116.11. Once you have a domain, you can make as many subdomains (www.mydomain.com
, irc.mydomain.com, i.like.to.use.leet.stuff.mydomain.com, ...) as you want and have those names point to wherever you want (your server would be a good place, but you could also have it point to your home computer's ip address, the address of the backup server of microsoft, the lan address of your gamecube, ...) This means you have an extra way to get to a server or computer instead of memorizing 4 numbers.
If you have one server running Apache, an ircd and a ftp server, you can make different dns names like irc.mydomain.com and www.mydomain.com
, but they all point to the same ip address 18.104.22.168, so you could surf to ftp.mydomain.com or ftp to irc.mydomain.com, it's all the same server.
For an irc server, you download an ircd, a program that serves irc like Apache would serve a web site, and you install and run that program. Then you can do /server 22.214.171.124 in mIRC, or you can assign your server a (another) dns name and use that.
So, yes, if you have a computer thats accessible from the internet, you can run an irc server.
Note I didn't say an irc network, a network requires multiple servers running the same irc server application and being connected to each other. For linking your server to an existing network, you'll have to ask the owners of that network and get no for an answer anyway.