Web design, development, and maintenance.
A web service allows a PROGRAM to talk to a web page, instead of using your browser to open a web page.
Example: I can go to maps.google.com, and type in my home address, and see a map of where I live in my browser.
But what if you were writing a computer program where you wanted to take an address and show a pretty map, just like Google maps?
Well, you could write a whole new mapping program from scratch, OR you could call a web service that Google maps provides, send it the address, and it will return a graphical map of the location, which you can display in your program.
There is a lot more to it, as some of the other posts go into, but the upshot is that it allows your application to either retrieve information FROM, or submit information TO some resource.