API's for other sites - how to integrate?

hi there - my apologies, i'm not quite sure which plugin this should be allocated to.

so i'm busy with a collaboration with a big international site. basically, what we're doing, is their folks can sign up on my website, and get x -amount discount on a product - but then they become members of my network, much like facebook integration would let you sign in? so,
it needs to be more than just an affiliate marketing thing.

i've started teh conversation with these guys to get an API key from them, and they do ask on their site why i would want it, and what the URL would be and stuff like that... to give you an idea of what they're doing with the application on their end.

if this was FB, i'd know how to "create an app", but it's possibly a bit more hairy than that to do this integration....

please point me in the right direction?
what plugins will i need for my solution on my end?
how do we integrate their stuffs with our stuffs?


*beams excitedly*

thanks for always helping!