If you want to secure your API in a server to server fashion (no redirection to website for 2 legged authentication). You can look at OAuth2 Client Credentials Grant protocol.
https://dev.twitter.com/docs/auth/application-only-auth
I have developed a library that can help you easily add this kind of support to your WebAPI. You can install it as a NuGet package:
https://nuget.org/packages/OAuth2ClientCredentialsGrant/1.0.0.0
The library targets .NET Framework 4.5.
Once you add the package to your project, it will create a readme file in the root of your project. You can look at that readme file to see how to configure/use this package.
Cheers!
