REST is a software design pattern typically used for web applications. In layman's terms this means that it is a commonly used idea used in many different projects. It stands for REpresentational State Transfer. The basic idea of REST is treating objects on the server-side (as in rows in a database table) as resources than can be created or destroyed.
The most basic way of thinking about REST is as a way of formatting the URLs of your web applications. For example, if your resource was called "posts", then:
/posts Would be how a user would access ALL the posts, for displaying.
/posts/:id Would be how a user would access and view an individual post, retrieved based on their unique id.
/posts/new Would be how you would display a form for creating a new post.
Sending a POST request to /users would be how you would actually create a new post on the database level.
Sending a PUT request to /users/:id would be how you would update the attributes of a given post, again identified by a unique id.
Sending a DELETE request to /users/:id would be how you would delete a given post, again identified by a unique id.
As I understand it, the REST pattern was mainly popularized (for web apps) by the Ruby on Rails framework, which puts a big emphasis on RESTful routes. I could be wrong about that though.
I may not be the most qualified to talk about it, but this is how I've learned it (specifically for Rails development).
When someone refers to a "REST api," generally what they mean is an api that uses RESTful urls for retrieving data.
