Web services are defined differently at different places. Word-by word translation states that any service provided on the web including a web page is a service but this isn't true if the technical term web service is referred to.
To define web service, we will look at web service definition from the W3C glossary:
"A Web service is a software system designed to support inter-operable machine-to-machine interaction over a network. It has an interface described in a machine-process able format (specifically WSDL). Other systems interact with the Web service in a manner prescribed by its description using SOAP-messages, typically conveyed using HTTP with an XML serialization in conjunction with other Web-related standards." -W3C, web services glossary.
This definition again, is not completely true as it is more specific to SOAP and WSDL based web services. In fact, later in the W3C Working Group Note, February 11, 2004, it was stated that:
"We can identify two major classes of web services:- REST-compliant web services, in which the primary purpose of the service is to manipulate XML representations of Web resources using a uniform set of "stateless" operations; - and arbitrary web services, in which the service may expose an arbitrary set of operations."
So considering that, a more general and better definition of a web service is this, from the preceding mentioned W3C web services glossary definition:
"A Web service is a software system designed to support inter-operable machine-to-machine interaction over a network."
Now, we know what a web service is. So proceeding to REST, it is important to know the need for a web service. Where can a web service be used?
As just defined, a web service is a system to machine-to-machine inter-operable communication over a network. It is very useful for communication between different systems or devices. In our case, we will be using web services to provide an interface by which either a mobile application or a web application will be able to communicate with a server to get and store data. This will make the client-side application separate from the server side logic. And nowadays, SPAs (Single Page Applications) and mobile applications need to be stand alone, separate from server side and only interacting with server side logic with web services. So definitely web services are very much important nowadays. However, web service usage is not limited to client side application usage but it is also useful in server to server communication where one server acts as a client.