WebServices are a way for programs to talk to each other via the Web. In essence, they're an HTTP-based (and thus more bloated and more trendy) reiteration of RPC. They are a rarity on the open web; the bonmot goes that “there are more SOAP implementations than there are WebServices on the open web.”
Probably the most well-known specimen is Amazon’s SOAP API, which offers a lot of information about books and ways to sell them, so you can display specific information about books available at Amazon on your own WebSite (perhaps listing all the books you authored). This information will remain in sync with the database at Amazon.