For example, check out this Facebook plugin.

In the client side the API key is clearly visible. What is stopping another user from obtaining this key and using this feature on a different site?

I figured a very naive implementation would be to check the domain the request comes from but things like this are easy to fake.

If I were to create something similar, how would I go about securing the authentication process?

I want as much of this work to be client side, though some form of server authentication will be required surely? Any links or advice would be greatly appreciated.

Update

Similar question about API keys that I found useful.