From here:

/**
*
*  Base64 encode / decode
*  http://www.webtoolkit.info/
*
**/
var Base64 = {

    // private property
    _keyStr : "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=",

    // public method for encoding
    encode : function (input) {
        var output = "";
        var chr1, chr2, chr3, enc1, enc2, enc3, enc4;
        var i = 0;

        input = Base64._utf8_encode(input);

        while (i < input.length) {

            chr1 = input.charCodeAt(i++);
            chr2 = input.charCodeAt(i++);
            chr3 = input.charCodeAt(i++);

            enc1 = chr1 >> 2;
            enc2 = ((chr1 & 3) << 4) | (chr2 >> 4);
            enc3 = ((chr2 & 15) << 2) | (chr3 >> 6);
            enc4 = chr3 & 63;

            if (isNaN(chr2)) {
                enc3 = enc4 = 64;
            } else if (isNaN(chr3)) {
                enc4 = 64;
            }

            output = output +
            this._keyStr.charAt(enc1) + this._keyStr.charAt(enc2) +
            this._keyStr.charAt(enc3) + this._keyStr.charAt(enc4);
        }
        return output;
    },

    // public method for decoding
    decode : function (input) {
        var output = "";
        var chr1, chr2, chr3;
        var enc1, enc2, enc3, enc4;
        var i = 0;

        input = input.replace(/[^A-Za-z0-9\+\/\=]/g, "");

        while (i < input.length) {

            enc1 = this._keyStr.indexOf(input.charAt(i++));
            enc2 = this._keyStr.indexOf(input.charAt(i++));
            enc3 = this._keyStr.indexOf(input.charAt(i++));
            enc4 = this._keyStr.indexOf(input.charAt(i++));

            chr1 = (enc1 << 2) | (enc2 >> 4);
            chr2 = ((enc2 & 15) << 4) | (enc3 >> 2);
            chr3 = ((enc3 & 3) << 6) | enc4;

            output = output + String.fromCharCode(chr1);

            if (enc3 != 64) {
                output = output + String.fromCharCode(chr2);
            }
            if (enc4 != 64) {
                output = output + String.fromCharCode(chr3);
            }
        }

        output = Base64._utf8_decode(output);

        return output;
    },

    // private method for UTF-8 encoding
    _utf8_encode : function (string) {
        string = string.replace(/\r\n/g,"\n");
        var utftext = "";

        for (var n = 0; n < string.length; n++) {

            var c = string.charCodeAt(n);

            if (c < 128) {
                utftext += String.fromCharCode(c);
            }
            else if((c > 127) && (c < 2048)) {
                utftext += String.fromCharCode((c >> 6) | 192);
                utftext += String.fromCharCode((c & 63) | 128);
            }
            else {
                utftext += String.fromCharCode((c >> 12) | 224);
                utftext += String.fromCharCode(((c >> 6) & 63) | 128);
                utftext += String.fromCharCode((c & 63) | 128);
            }
        }
        return utftext;
    },

    // private method for UTF-8 decoding
    _utf8_decode : function (utftext) {
        var string = "";
        var i = 0;
        var c = c1 = c2 = 0;

        while ( i < utftext.length ) {

            c = utftext.charCodeAt(i);

            if (c < 128) {
                string += String.fromCharCode(c);
                i++;
            }
            else if((c > 191) && (c < 224)) {
                c2 = utftext.charCodeAt(i+1);
                string += String.fromCharCode(((c & 31) << 6) | (c2 & 63));
                i += 2;
            }
            else {
                c2 = utftext.charCodeAt(i+1);
                c3 = utftext.charCodeAt(i+2);
                string += String.fromCharCode(((c & 15) << 12) | ((c2 & 63) << 6) | (c3 & 63));
                i += 3;
            }
        }
        return string;
    }
}

Also, search for "JavaScript base64 encoding" turns up a lot of other options, and the above was the first one.

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