using brackets with javascript import syntax

import React, { Component, PropTypes } from 'react';

This says:

Import the default export from 'react' under the name React and import the named exports Component and PropTypes under the same names.

This combines the two common syntaxes which you've probably seen

import React from 'react';
import { Component, PropTypes } from 'react';

The first being used to import and name the default export, the second to import the specified named exports.

As a general rule, most modules will either provide a single, default export, or a list of named exports. It is somewhat less usual for a module to provide both a default export and named exports. However, in the case where there is one feature which is most commonly imported, but also additional sub-features, it is a valid design to export the first as the default, and the remaining ones as named exports. It is in such cases you would use the import syntax you refer to.

The other answers are somewhere between wrong and confusing, possibly because the MDN documents at the time this question was asked were wrong and confusing. MDN showed the example

import name from "module-name";

and said name is the "name of the object that will receive the imported values." But that's misleading and incorrect; first of all, there is only one import value, which will be "received" (why not just say "assigned to", or "used to refer to") name, and the import value in this case is the default export from the module.

Another way of explaining this is to note that the above import is precisely identical to

import { default as name } from "module-name";

and the OP's example is precisely identical to

import { default as React, Component, PropTypes } from 'react';

The MDN documentation went on to show the example

import MyModule, {foo, bar} from "my-module.js";

and claimed that it means

Import an entire module's contents, with some also being explicitly named. This inserts myModule (sic), foo, and bar into the current scope. Note that foo and myModule.foo are the same, as are bar and myModule.bar

What MDN said here, and what other answers claim based on the incorrect MDN documentation, is absolutely wrong, and may be based on an earlier version of the spec. What this actually does is

Import the default module export and some explictly named exports. This inserts MyModule, foo, and bar into the current scope. The export names foo and bar are not accessible via MyModule, which is the default export, not some umbrella covering all exports.

(The default module export is the value exported with the export default syntax, which could also be export {foo as default}.)

The MDN documentation writers may have gotten confused with the following form:

import * as MyModule from 'my-module';

This imports all exports from my-module, and makes them accessible under names such as MyModule.name. The default export is also accessible as MyModule.default, since the default export is really nothing more than another named export with the name default. In this syntax, there is no way to import only a subset of the named exports, although one could import the default export, if there is one, together with all the named exports, with

import myModuleDefault, * as myModule from 'my-module';