Altered piece of code I often use:

Date.prototype.yyyymmdd = function() {
  var mm = this.getMonth() + 1; // getMonth() is zero-based
  var dd = this.getDate();

  return [this.getFullYear(),
          (mm>9 ? '' : '0') + mm,
          (dd>9 ? '' : '0') + dd
         ].join('');
};

var date = new Date();
date.yyyymmdd();

The best string format for string parsing is the date ISO format together with the JavaScript Date object constructor.

Examples of ISO format: YYYY-MM-DD or YYYY-MM-DDTHH:MM:SS.

But wait! Just using the "ISO format" doesn't work reliably by itself. String are sometimes parsed as UTC and sometimes as localtime (based on browser vendor and version). The best practice should always be to store dates as UTC and make computations as UTC.

To parse a date as UTC, append a Z - e.g.: new Date('2011-04-11T10:20:30Z').

To display a date in UTC, use .toUTCString(),
to display a date in user's local time, use .toString().

More info on MDN | Date and this answer.

For old Internet Explorer compatibility (IE versions less than 9 do not support ISO format in Date constructor), you should split datetime string representation to it's parts and then you can use constructor using datetime parts, e.g.: new Date('2011', '04' - 1, '11', '11', '51', '00'). Note that the number of the month must be 1 less.

Alternate method - use an appropriate library:

You can also take advantage of the library Moment.js that allows parsing date with the specified time zone.