I believe that Wikipedia provides a very nice explanation:
https://en.wikipedia.org/wiki/WebSocket
Just as it says, WebSocket is a mechanism whereby a user-agent (typically a browser) uses a raw TCP data channel for communication with the server, rather than encapsulating the data in HTTP messages (though the channel is initiated via an HTTP Request and established after an appropriate HTTP Response).
The BIG-IP supports WebSocket traffic:
https://support.f5.com/kb/en-us/solutions/public/14000/700/sol14754.html
Indeed, even though the initial handshake is HTTP and subsequent data are not, if the Virtual Server has the tcp profile but not the http profile, the traffic will work as expected in any version of BIG-IP (assuming you expect that the traffic is all handled as a single, load-balanced TCP flow).
If you want to use the http profile, that'll work fine starting with 11.4.0. Before that, you need to employ an iRule:
https://support.f5.com/kb/en-us/solutions/public/14000/800/sol14814.html