A shell is basically a user account on a computer somewhere that's connected 24/7. Most commonly people use a Linux shell that runs Linux software, but there are some Windows shells out there that could run mIRC.
Most bots are written for Linux, so hosting mIRC as a bot remotely isn't going to be as straight forward as simply installing a ready-made bot for Linux.
Of course, you could simply leave your computer powered on and connected to the Internet 24/7.
A BNC (common software used is ZNC) is just a proxy relay for a client to connect through, except when your client disconnects the proxy will remain connected and await your return. Your bot wouldn't be responsive for the duration that mIRC is disconnected from the BNC.
