WCF stands for Windows Communication Foundation, created by Microsoft with .NET Framework 3.0. It provides us a platform where we can build secure, reliable service that can integrate across the platforms and interoperate smoothly.
Web API is also a framework that is used to make HTTP services. As you know, now a days we are using mobiles, tablets, apps and different types of services, so Web API is a simple and reliable platform to create HTTP enabled services that can reach wide range of clients. It is used to create complete REST services.
RESTful services like Web API are simple and light weight and easy to access by any client which supports HTTP calls. It is not like to SOAP where client need to understand WSDL to implement the service.
WSDL uses some extra configuration and require some additional logic that might not be available on many devices such as phones, smart TVs, Tablets, etc.
As WCF supports many protocol so it can consumed by lots of client that are using different protocol such as TCP, MSMQ, etc. not only HTTP protocol to access service.
For more information about WCF and Web API please refer below link from Stack Overflow
Difference between WCF and Web API is given below: