Many organizations use multiple software systems for management. Different software systems often need to exchange data with each other, and a Web service is a method of communication that allows two software systems to exchange this data over the Internet. The software system that requests data is called a service requester, whereas the software system that would process the request and provide the data is called a service provider.
A web service is something that a website chooses to offer to those who wish to read, update and/or delete data from your website. You might call it a "backdoor" to your data. Instead of presenting the data as part of a webpage it is provided in a pre-determined way where some of the more popular are XML and JSON. There are several ways to communicate with a webservice, some use SOAP, others have REST'ful web services, etc.
A web service always hides the internal complexity of the service to its clients. For an example, an airliner which is selling tickets via a third party portal is only gathering end user info via a web service and in return it is providing ticket. The portal which is selling ticket doesn't have to bother about the internal complexity of the airlines ticket reservation systems.