Distributed Application

A distributed application is defined as software that runs on two or more computers. Designing a distributed application in ASP. NET MVC is similar in that you have the client (or view) in the browser, the business layer (the model), and the data access layer behind the model.

Web Services

Web Service is an abstract term encompassing a large variety of data providers for distributed systems. Perhaps you are referring to ASMX web services, which can still be found in the wild but aren't really widely used in new development these days.

WCF Service is Microsoft's implementation of SOAP. There are others implementations or you could roll your own (not recommended).

SOAP is a kind of stateful, session-based, message-based web service. It's good if your service is designed as a set of complex actions.

REST is a stateless, sessionless, resource-based web service. It's good if your service is designed to access data and perform simple CRUD operations on it. SOAP and REST are mutually exclusive. A service cannot be both. There are ways to manipulate vanilla WCF to make is RESTful but these techniques are becoming deprecated. If you want to implement a RESTful web service there are two main choices in the Microsoft world: WCF Data Services and ASP.NET Web API.

ASMX file

ASMX files was the older standard for creating web services using SOAP structure. It is now obsolete and the recommended web services to use are WCF and Web API. WCF using SOAP structure while Web API uses REST.