I originally wrote this answer when TypeScript was still hot-off-the-presses. Five years later, this is an OK overview, but look at Lodewijk's answer below for more depth

1000ft view...

TypeScript is a superset of JavaScript which primarily provides optional static typing, classes and interfaces. One of the big benefits is to enable IDEs to provide a richer environment for spotting common errors as you type the code.

To get an idea of what I mean, watch Microsoft's introductory video on the language.

For a large JavaScript project, adopting TypeScript might result in more robust software, while still being deployable where a regular JavaScript application would run.

It is open source, but you only get the clever Intellisense as you type if you use a supported IDE. Initially, this was only Microsoft's Visual Studio (also noted in blog post from Miguel de Icaza). These days, other IDEs offer TypeScript support too.

Are there other technologies like it?

There's CoffeeScript, but that really serves a different purpose. IMHO, CoffeeScript provides readability for humans, but TypeScript also provides deep readability for tools through its optional static typing (see this recent blog post for a little more critique). There's also Dart but that's a full on replacement for JavaScript (though it can produce JavaScript code)

Example

As an example, here's some TypeScript (you can play with this in the TypeScript Playground)

class Greeter {
    greeting: string;
    constructor (message: string) {
        this.greeting = message;
    }
    greet() {
        return "Hello, " + this.greeting;
    }
}

And here's the JavaScript it would produce

var Greeter = (function () {
    function Greeter(message) {
        this.greeting = message;
    }
    Greeter.prototype.greet = function () {
        return "Hello, " + this.greeting;
    };
    return Greeter;
})();

Notice how the TypeScript defines the type of member variables and class method parameters. This is removed when translating to JavaScript, but used by the IDE and compiler to spot errors, like passing a numeric type to the constructor.

It's also capable of inferring types which aren't explicitly declared, for example, it would determine the greet() method returns a string.

Debugging TypeScript

Many browsers and IDEs offer direct debugging support through sourcemaps. See this Stack Overflow question for more details: Debugging TypeScript code with Visual Studio

Want to know more?

I originally wrote this answer when TypeScript was still hot-off-the-presses. Check out Lodewijk's answer to this question for some more current detail.