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.
Web services are a type of internet software that use standardized messaging protocols and are made available from an application service provider’s web server for use by a client or other web-based programs. Web services can range from major services such as storage management or customer relationship management (CRM) down to much more limited services such as the furnishing of a stock quote or the checking of bids for an auction item. The term is sometimes also referred to as application services.