ASP

Application Service Provider. see also SaaS

This "industry"/category has been defined so broadly as to be meaningless. Some companies basically license 3rd party software (e.g. MsExchange) and host it for you. I'm more interested in companies that build their own software in-house and then host it for you.

Paul Graham has written about the joys of server-based software.

Also ActiveServerPages, name of pre-DotNet MsWindows Web Development platform

http://en.wikipedia.org/wiki/Active_Server_Pages