All ASP.NET web frameworks are build on top of Microsoft ASP.NET Framework. The unique feature of Web API is that it can be used with both MVC and WebForms applications to provide truly restful HTTP services.
Regarding the choice of suitable Asp.NET framework: you may get more information from official source - www.asp.net.

General rule of thumb is the architectural design how you want to build your application.
ASP.NET MVC promotes a cleaner separation which makes the developer think more in depth about design and code separation than traditional web forms.
There are endless debates about what is better but true benefits of ASP.NET MVC as
- testability
- more control over the rendered HTML
- separation of concerns. However with MVC there is much more to learn for the developer.
ASP.NET WebForms - will always be around because some see it as a rapid application development tool. Just drag and drop and let ASP.NET handle the posting, state etc
ASP.NET Web API - is a framework that makes it easy to build HTTP services that reach a broad range of clients, including browsers and mobile devices. ASP.NET Web API is an ideal platform for building RESTful applications on the .NET Framework.
All in all, each of them has certain business solutions to be considered as required.
References to look for comparison: