A client is anything connecting to a server that is not another server.
– J. Oikarinen, D. Reed; Internet Relay Chat Protocol; RFC 1459; May 1993.
When you log onto IRC with your client application, it logs onto the server you set as your default in the program settings. You can then join any public chat room running on that server. Every message you enter in a given chat room is then sent by your client application across the Internet to the IRC server, which immediately echoes it to every other user connected to that chat room.
Because the server copies and echoes the messages across the world so fast, traveling across copper and fiber connections at 2/3 the speed of light, the illusion is maintained that everyone is directly connected to everyone else, even though they are only really connected in common to the server.