This changed a bit in babel v6.

From the docs:

The polyfill will emulate a full ES6 environment. This polyfill is automatically loaded when using babel-node.

Installation:
$ npm install babel-polyfill

Usage in Node / Browserify / Webpack:
To include the polyfill you need to require it at the top of the entry point to your application.
require("babel-polyfill");

Usage in Browser:
Available from the dist/polyfill.js file within a babel-polyfill npm release. This needs to be included before all your compiled Babel code. You can either prepend it to your compiled code or include it in a <script> before it.

NOTE: Do not require this via browserify etc, use babel-polyfill.

The Babel docs describe this pretty concisely:

Babel includes a polyfill that includes a custom regenerator runtime and core.js.

This will emulate a full ES6 environment. This polyfill is automatically loaded when using babel-node and babel/register.

Make sure you require it at the entry-point to your application, before anything else is called. If you're using a tool like webpack, that becomes pretty simple (you can tell webpack to include it in the bundle).

If you're using a tool like gulp-babel or babel-loader, you need to also install the babel package itself to use the polyfill.

Also note that for modules that affect the global scope (polyfills and the like), you can use a terse import to avoid having unused variables in your module:

import 'babel/polyfill';

For Babel version 7, if your are using @babel/preset-env, to include polyfill all you have to do is add a flag 'useBuiltIns' with the value of 'usage' in your babel configuration. There is no need to require or import polyfill at the entry point of your App.

With this flag specified, babel@7 will optimize and only include the polyfills you needs.

To use this flag, after installation:

npm install --save-dev  @babel/core @babel/cli @babel/preset-env
npm install --save @babel/polyfill

Simply add the flag:

useBuiltIns: "usage" 

to your babel configuration file called "babel.config.js" (also new to Babel@7), under the "@babel/env" section:

// file: babel.config.js

module.exports = () => {
   const presets = [
      [
         "@babel/env", 
         { 
             targets: { /* your targeted browser */ },
             useBuiltIns: "usage"  // <-----------------*** add this
         }
      ]
   ];

   return { presets };
};

Reference:

• usage#polyfill
• babel-polyfill#usage-in-node-browserify-webpack
• babel-preset-env#usebuiltins

Update Aug 2019:

With the release of Babel 7.4.0 (March 19, 2019) @babel/polyfill is deprecated. Instead of installing @babe/polyfill, you will install core-js:

npm install --save core-js@3

A new entry corejs is added to your babel.config.js

// file: babel.config.js

module.exports = () => {
   const presets = [
      [
         "@babel/env", 
         { 
             targets: { /* your targeted browser */ },
             useBuiltIns: "usage",
             corejs: 3  // <----- specify version of corejs used
         }
      ]
   ];

   return { presets };
};

see example: https://github.com/ApolloTang/stackoverflow-eg--babel-v7.4.0-polyfill-w-core-v3

Reference:

• 7.4.0 Released: core-js 3, static private methods and partial application
• core-js@3, babel and a look into the future