Hello TinVin727,
Depending on who you talk to, you will probably get different answers as these are loaded terms depending someone's background and opinion.
Web Service - Out of
Webopedia, MSDN
and
Wikipedia, I would say Wikipedia has the closest definition to how I think of web services:
"... a method of communication between two electronic devices over a network."
Web API is a Microsoft framework for creating web services.
Take a look at this post: http://stackoverflow.com/questions/19336347/what-is-the-difference-between-a-web-api-and-a-web-service
Note that an API is different from Web API. This is also described in the post above.
Cheers, Jeff