Not really a server you would register on but a network. What IRC network do you currently frequent? Dalnet, EFnet, Undernet, etc.. Many networks are more popular in different ports of the world.
Are you really looking for a IRC network to use, you could even use more than 1 - or just some real time chat on your website? There are many ways to accomplish that. Are you going to be the only help, are there going to be 100 or 1000s of people in the chat.
I host a chat for game I play, along with forums for the same guild, etc. If you have a database backend on your site its very easy to host multiple chat channels and private chats on the server if your not talking 1000's of concurrent users, etc.
There are services some free that can host realtime chat for you from your site where you need do nothing more than place some code on your site for users to click and access the chat.
Running a actual IRC channel is going to come with its own learning curve - prob going to want a bot for the channel, and then somewhere to host said bot, etc.
what exactly are you looking to accomplish the irc channel and be more likely to get multiple methods to accomplish it. If you just want to register a channel on a irc network, you can create a channel on any network - and if you start the channel you will have op control over said channel. If you looking to register a channel so its always yours, then you could use something like chanserv bot that is on many irc networks, or efnet has chanfix that can get you back ops on a channel you created but that gets taken over, etc.
Running a channel on IRC is bit more complicated than say registering a domain
And to be honest IRC is not really as popular as it was back in the day - not sure its what your looking for exactly. I have not played with it in many years, but use to run my own channel back in the day.. So sure I could get you started, etc. But lets get to the heart of what you want exactly and might find an easier route to accomplish that goal.