Those two lines of code simply trigger a change event to this.element using two different allowed syntax.
Using .trigger():
this.element.trigger('change');
Or using a shorthand method .change():
that.element.change();
You can actually bind an event handler to the element represented by this.element to handle this event.
Without knowing your plugin, it is difficult to answer you precisely on what is this.element.
But take the example of the autocomplete plugin. In this one, this.element is actually the input field the autocomplete plugin is applied to. If the change event was triggered like supposedly done in your question, you could bind an event handler to the input like this:
$('#myinput')
.autocomplete()
.bind('change', function() { });
Now if this plugin relies on the jQuery UI Widget Factory, it is advisable to use the method _trigger() to trigger events instead of the jquery .trigger().
Using _trigger() will make sure to execute any callback defined in the plugin's option with a correct context and also trigger that event for this.element (like above). So you could have:
$('#myinput')
.somePlugin({
change: function(e, someData) {
// "this" here will be `this.element´
}
})
.bind('change', function() { ... });