I'm designing a public API to my company's data. We want application developers to sign up for an API key so that we can monitor use and overuse.
Since the API is REST, my initial thought is to put this key in a custom header. This is how I've seen Google, Amazon, and Yahoo do it. My boss, on the other hand, thinks the API is easier to use if the key becomes merely a part of the URL, etc. "http://api.domain.tld/longapikey1234/resource". I guess there is something to be said for that, but it violates the principle of the URL as a simple address of what you want, and not how or why you want it.
It should be put in the HTTP Authorization header. The spec is here https://www.rfc-editor.org/rfc/rfc7235