What is REST?
REST is a term coined by Roy Fielding in his Ph.D. dissertation  to describe an architecture style of networked systems. REST is an acronym standing for Representational State Transfer.
Why is it called Representational State Transfer?
The Web is comprised of resources. A resource is any item of interest. For example, the Boeing Aircraft Corp may define a 747 resource. Clients may access that resource with this URL:
A representation of the resource is returned (e.g., Boeing747.html). The representation places the client application in a state. The result of the client traversing a hyperlink in Boeing747.html is another resource is accessed. The new representation places the client application into yet another state. Thus, the client application changes (transfers) state with each resource representation --> Representational State Transfer!
Here is Roy Fielding's explanation of the meaning of Representational State Transfer:
"Representational State Transfer is intended to evoke an image of how a well-designed Web application behaves: a network of web pages (a virtual state-machine), where the user progresses through an application by selecting links (state transitions), resulting in the next page (representing the next state of the application) being transferred to the user and rendered for their use."
Motivation for REST
The motivation for REST was to capture the characteristics of the Web which made the Web successful. Subsequently these characteristics are being used to guide the evolution of the Web.
REST - An Architectural Style, Not a Standard
REST is not a standard. You will not see the W3C putting out a REST specification. You will not see IBM or Microsoft or Sun selling a REST developer's toolkit. Why? Because REST is just an architectural style. You can't bottle up that style. You can only understand it, and design your Web services in that style. (Analogous to the client-server architectural style. There is no client-server standard.)