It's a framework to build ambitious web applications. It's a client side framework meaning that it runs in a web browser. That means the application does not have to reload a full page, it can just fetch data it needs and update only components that have to be re-rendered.
It's a opinionated framework meaning that instead of configuring it you follow already set conventions. This means that when you'll understand one application written in Ember, you'll be able to jump into another app and find your way very quickly.
We started to use Ember about a year ago when we started a major rewrite of our mobile site. We aimed to build a skin that will be agnostic from data source and communicates with server only via APIs and Ember is a perfect tool for this job.
You can check how we use Ember at Wikia in this repository: https://github.com/Wikia/mercury