Web services are services exposed over a network and is designed to be used automatically by other applications without human interaction.
A practical example for web services is a news web site which also shows cricket score. The cricket score service can be provided as a web service on a different server and our news web site can automatically invoke that service without any human intervention, get the result and update it along with the news. Note that this can also be done using Ajax.
Web services are different from web applications. While web applications are applications with web pages that reside on a server, primarily for use by humans, a web service is a server-based application component meant primarily for interaction with other programs. A web application might use a web service to retrieve and populate some data from another server without the end user knowing about it.
There are two types of web services: SOAP web services and REST based webservices. While SOAP is based on a set of predefined rules, REST is a newer and popular model which is based on guidelines and existing protocols.