If you are getting a JS based date String then first use the new Date(String) constructor and then pass the Date object to the moment method. Like:
var dateString = 'Thu Jul 15 2016 19:31:44 GMT+0200 (CEST)';
var dateObj = new Date(dateString);
var momentObj = moment(dateObj);
var momentString = momentObj.format('YYYY-MM-DD'); // 2016-07-15
In case dateString is 15-07-2016, then you should use the moment(date:String, format:String) method
var dateString = '07-15-2016';
var momentObj = moment(dateString, 'MM-DD-YYYY');
var momentString = momentObj.format('YYYY-MM-DD'); // 2016-07-15