Say I create an object as follows:

var myObject = {
    "ircEvent": "PRIVMSG",
    "method": "newURI",
    "regex": "^http://.*"
};

What is the best way to remove the property regex to end up with new myObject as follows?

var myObject = {
    "ircEvent": "PRIVMSG",
    "method": "newURI"
};

var test = {'red':'#FF0000', 'blue':'#0000FF'};
delete test.blue; // or use => delete test['blue'];
console.log(test);

this deletes test.blue

There are two ways to add new properties to an object:

var obj = {
    key1: value1,
    key2: value2
};

Using dot notation:

obj.key3 = "value3";

Using square bracket notation:

obj["key3"] = "value3";

The first form is used when you know the name of the property. The second form is used when the name of the property is dynamically determined. Like in this example:

var getProperty = function (propertyName) {
    return obj[propertyName];
};

getProperty("key1");
getProperty("key2");
getProperty("key3");

A real JavaScript array can be constructed using either:

The Array literal notation:

var arr = [];

The Array constructor notation:

var arr = new Array();

Your element is not an array, however your cart needs to be an array in order to support many element objects. Code example:

var element = {}, cart = [];
element.id = id;
element.quantity = quantity;
cart.push(element);

If you want cart to be an array of objects in the form { element: { id: 10, quantity: 1} } then perform:

var element = {}, cart = [];
element.id = id;
element.quantity = quantity;
cart.push({element: element});

JSON.stringify() was mentioned as a concern in the comment:

>> JSON.stringify([{a: 1}, {a: 2}]) 
      "[{"a":1},{"a":2}]" 

You’ll have to iterate:

for (var k in obj) {
  if (!obj.hasOwnProperty(k)) continue;
  if (obj[k] === "orange") {
    /* yaay! an orange! */
  }
}

Now that “hasOwnProperty” test in there is to make sure you don’t stumble over properties inherited from the prototype. That might not be desirable in some cases, and really it’s one of the things you kind-of need to understand in order to know for sure whether you do or don’t want to make that test. Properties that you pick up from an object’s prototype can sometimes be things dropped there by various libraries. (I think the ES5 standard provides for ways to control whether such properties are “iterable”, but in the real world there’s still IE7.)