How to detect a textbox's content has changed

Start observing 'input' event instead of 'change'.

jQuery('#some_text_box').on('input', function() {
    // do your stuff
});

...which is nice and clean, but may be extended further to:

jQuery('#some_text_box').on('input propertychange paste', function() {
    // do your stuff
});

Use the onchange event in HTML/standard JavaScript.

In jQuery that is the change() event. For example:

$('element').change(function() { // do something } );

EDIT

After reading some comments, what about:

$(function() {
    var content = $('#myContent').val();

    $('#myContent').keyup(function() { 
        if ($('#myContent').val() != content) {
            content = $('#myContent').val();
            alert('Content has been changed');
        }
    });
});

The 'change' event doesn't work correctly, but the 'input' is perfect.

$('#your_textbox').bind('input', function() {
    /* This will be fired every time, when textbox's value changes. */
} );