A Web service is commonly used to exchange datas between two or more network applications.

Personally I use it when a client want interact between his Odoo and an other application like an hight performance frontend or to import or export datas, like migration version, or migrate from another ERP to Odoo or from Odoo to an other ERP.

You can see a full example at : http://thierry-godin.developpez.com/openerp/openerp-xmlrpc-php-en/

A Web Service API is a communication protocol with external applications Odoo. Here you can find the documentation:

https://www.odoo.com/documentation/8.0/api_integration.html