Web service is a term that refers to the method of communication between two electronic devices over the Internet. It is an encapsulated logic (functionality) of an application that is used by another one. It is therefore a machine interaction - web services are NOT designed for a human interaction (such as browsing in the web browser).
Exposed web service has a defined public interface and functionality (over WSDL language) that are known by the other application in advance. Web service therefore allows an interaction of two applications, usually using XML language by the SOAP Protocol (over HTTP, SMTP, etc). There are also other ways of communication, such as using REST protocol.