Windows Communication Foundation is designed to exchange standard SOAP-based messages using a variety of transport protocols like HTTP, TCP, NamedPipes or MSMQ, etc. WCF i.e. Windows Communication Foundation is a framework used for building Service Oriented applications (SOA) and supports multiple transport protocols like HTTP, TCP, MSMQ, etc. WCF ships out with the .NET Framework. WCF service is good for Message Queue, duplex communication, one-way messaging. WCF supports SOAP and XML formats. 

ASP.NET API is a framework for building non-SOAP based services over HTTP only.  Web API is a Framework to build HTTP Services that can reach a board of clients, including browsers, mobile, IoT Devices, etc. and PROVIDED an ideal platform for building RESTful applications. It is LIMITED to HTTP based services. ASP.NET framework ships out with the .NET framework and is Open Source. One major advantage of Web API is that it is the best fit with MVC PATTERN. Web API supports any media format including JSON, XML. 

• Both Web API and WCF can be self-hosted or can be hosted on the IIS Server.