As of 2016, you can do this more simply than in any of the answers already given:
$('#myChosenBox').val();
where "myChosenBox" is the id of the original select input. Or, in the change event:
$('#myChosenBox').on('change', function(e, params) {
alert(e.target.value); // OR
alert(this.value); // OR
alert(params.selected); // also in Panagiotis Kousaris' answer
}
In the Chosen doc, in the section near the bottom of the page on triggering events, it says "Chosen triggers a number of standard and custom events on the original select field." One of those standard events is the change event, so you can use it in the same way as you would with a standard select input. You don't have to mess around with using Chosen's applied classes as selectors if you don't want to. (For the change event, that is. Other events are often a different matter.)