No, it's a full-stack framework, using Node.js on the server and sending down a client app bundle containing all HTML templates and JS. The client and server then communicate exclusively with JSON through a purpose-invented pubsub WebSockets/SockJS protocol they call DDP. The client has a simulated MongoDB database and thus much of the API is isomorphic, so the client will simulate the server's code while it awaits the actual server response. Client changes to the data are immediately effected in the browser DB, while the change is authorized on the server, synced to other server instances with MongoDB oplog, and then pushed down to all clients subscribed to that data. The templating engine is then reactively informed of the data change, and the DOM is updated at the lowest level possible such as modifying an individual text node.
You need to be running the Node.js Meteor application. The client relies on DDP rather than REST and right now Meteor is the only server framework using DDP. There is, however, an isomorphic HTTP library which makes it easy to communicate with third party REST APIs on both the client and server. It is also easy to create REST endpoints for your Meteor collections, so that third party services which don't support DDP can communicate with your app.
It is very possible to use other client side frameworks such as Angular or Polymer, rather than just using Meteor's Blaze UI engine. However I find that Blaze's Template<--Helper<--Data<--Event system is extremely easy to understand and reason about.
Edit: Actually, they do seem to have a decoupled version of the Blaze UI that you could use just for your reactive DOM code and hook into a REST API. You just won't get any of the server-dependent features like full-stack automated data synchronization, hot-reload, live CSS injection, or Meteor's build chain. You just get a simple reactive template/helper/events system. http://meteor.github.io/blaze/
It seems like too much magic until you've been exposed to it a bit. It's actually pretty simple to visualize what's going on under the hood, but it takes a bit to click. Reading the DDP spec and the tracker package definitely helped me grasp the concepts. The new subprojects page is also great at explaining the major Meteor components: https://www.meteor.com/projects
