This is a client command which tells your IRC client to send everything in this command directly to the server and let the server handle it. This command is called RAW on some clients. For those clients, simply substitute the word quote for raw in this article. Going forth, this article will use the word quote exclusively, including in command examples.
There are several reasons for using quote. Most commonly, there is an IRC server command which shares the same name as an IRC command in your client, but these two commands may do different things, or work differently in some way. One example is the cycle command found in Hexchat/xchat. The client command here will cause you to part and then attempt to rejoin the channel. This attempted rejoin can fail for various reasons, such as the channel being invite only. The server side cycle command will attempt to overcome these limitations. To use that one specifically though, you must use quote. To do a server side cycle of #channel, you would use this command:
/quote cycle #channel
More to come later.