Well, I'll try.
ASIO stands for "Audio Stream In/Out" and is a protocol for audio drivers developed by Steinberg. It's main advantage is very low latency to minimise the problems we hear about so often of lag between input and the monitoring of output when you're trying to monitor what you record.
Not all sound device, especially cheap built in sound cards, can handle ASIO though. Most use Windows WDM drivers which are okay for simple Youtube playback and such but fairly useless for any serious recording.
ASIO4ALL is a very useful little bit of software which lets you pretend that devices with WDM drivers are working under ASIO. ASIO4ALL doesn't mess with or change your WDM drivers, it just works around them.
"WaveOut" and "Directsound" are other APIs (Application Programming Interfaces) letting Windows Operating systems play digital sounds...WaveOut is the oldest and pretty limited in facilities with Directsound being a slightly more modern one, supporting multiple streams and hardware acceleration. However, these again are aimed more at "basic" computer tasks and gaming, not serious recording and mixing.
Dummy out I have no idea about!
Anyhow, to answer your question, for home studio working, I'd stick to ASIO if you sound card supports it or ASIO4ALL if it doesn't. (And, if it doesn't, I'd be budgeting for a better sound card as your next upgrade.)
Bob