I would suggest starting with the most straightforward solutions first - maybe simple HTTP Basic Authentication + HTTPS is enough in your scenario.
If not (for example you cannot use https, or need more complex key management), you may have a look at HMAC-based solutions as suggested by others. A good example of such API would be Amazon S3 (http://s3.amazonaws.com/doc/s3-developer-guide/RESTAuthentication.html)
I wrote a blog post about HMAC based authentication in ASP.NET Web API. It discusses both Web API service and Web API client and the code is available on bitbucket. http://www.piotrwalat.net/hmac-authentication-in-asp-net-web-api/
Here is a post about Basic Authentication in Web API: http://www.piotrwalat.net/basic-http-authentication-in-asp-net-web-api-using-message-handlers/
Remember that if you are going to provide an API to 3rd parties, you will also most likely be responsible for delivering client libraries. Basic authentication has a significant advantage here as it is supported on most programming platforms out of the box. HMAC, on the other hand, is not that standardized and will require custom implementation. These should be relatively straightforward but still require work.
PS. There is also an option to use HTTPS + certificates. http://www.piotrwalat.net/client-certificate-authentication-in-asp-net-web-api-and-windows-store-apps/
