Internet Explorer 10+

// Define the string
var string = 'Hello World!';

// Encode the String
var encodedString = btoa(string);
console.log(encodedString); // Outputs: "SGVsbG8gV29ybGQh"

// Decode the String
var decodedString = atob(encodedString);
console.log(decodedString); // Outputs: "Hello World!"

Cross-Browser

Re-written and modularized UTF-8 and Base64 Javascript Encoding and Decoding Libraries / Modules for AMD, CommonJS, Nodejs and Browsers. Cross-browser compatible.

with Node.js

Here is how you encode normal text to base64 in Node.js:

//Buffer() requires a number, array or string as the first parameter, and an optional encoding type as the second parameter. 
// Default is utf8, possible encoding types are ascii, utf8, ucs2, base64, binary, and hex
var b = Buffer.from('JavaScript');
// If we don't use toString(), JavaScript assumes we want to convert the object to utf8.
// We can make it convert to other formats by passing the encoding type to toString().
var s = b.toString('base64');

And here is how you decode base64 encoded strings:

var b = Buffer.from('SmF2YVNjcmlwdA==', 'base64')
var s = b.toString();

with Dojo.js

To encode an array of bytes using dojox.encoding.base64:

var str = dojox.encoding.base64.encode(myByteArray);

To decode a base64-encoded string:

var bytes = dojox.encoding.base64.decode(str)

bower install angular-base64

<script src="bower_components/angular-base64/angular-base64.js"></script>

angular
    .module('myApp', ['base64'])
    .controller('myController', [

    '$base64', '$scope', 
    function($base64, $scope) {
    
        $scope.encoded = $base64.encode('a string');
        $scope.decoded = $base64.decode('YSBzdHJpbmc=');
}]);

But How?

If you would like to learn more about how base64 is encoded in general, and in JavaScript in-particular, I would recommend this article: Computer science in JavaScript: Base64 encoding