A web service allows programs and websites to talk to each other, in a simple phrase: machine-to-machine communication. Web services can be used to send website data to an outside application, implement 3rd party functionality or back-end services on a website.
For example, if I was building an application and wanted to display a custom map of the user's location, I could build an entire map application from scratch OR I can use the web service of an existing map application, like Google Maps. I could send the user's address to the Google Map's web service, and it would return a graphical map of the location which I can customize and display.
At their core, web services allow you to receive and submit information to other programs and websites, without navigating to them directly. You can submit orders, research things, send out information about your company and product, allow for applications to access your site, etc... This is a flow of information that can be built to go either or both ways, to your own benefit and those of sites and applications your site would interact with.
