BNC = Bouncer.
IRC bouncers connect to a server and stay connected at all times, preventing people from stealing your nickname, recording your messages, etc.
Some also have scripting capabilities built in, so you can use it to run scripts on channels and annoy people with it.
Most of all, they let you have a cool vhost (the hostname shown when you /whois somebody) - so, for example, I could change mine from "firstname.lastname@example.org" to "email@example.com" - which is, sadly, why most people get them.
Bouncers are in no way associated with a specific IRC client, so you should be able to get a BNC then connect through it with any client.
IMO, this isn't all that great. It has a few more features, and a better interface than mIRC (and it's free), but mIRC has more powerful features for real IRC users, and it has it's epic scripting language.