Australian Secret Intelligence here...
Our cover story is that ASIO stands for "Audio Stream Input Output". Rather than being specific to any soundcard or type of soundcard, it's a protocol developed by Steinberg that can be used by many different brands of hardware interfaces. The idea behind ASIO is to provide low latency and high quality.
The other main subset of drivers is generally Windows Direct drivers (usually referred to as WDM). These are the standard audio drivers used in Windows computers and tend to follow an intermediary path in the computer, adding to latency.
ASIO4ALL is a clever bit of software that can emulate ASIO in the operation of WDM devices. It's not strictly true that WDM drivers are only used in on-board gamers cards--a lot of hardware manufacturers offer both WDM and ASIO drivers even for high quality specialist cards.
What is right is that you're generally better off using a manufacturers own ASIO drivers if at all possible. However, sometimes there can be problems with compatibility in manufacturers drivers and certain OS's or DAW software. If that's the case, ASIO4ALL can be a handy fix, even with good cards. However, it's a fix and the manufacturers ASIO drivers are your best bet if they work properly.
Bob