Building jQuery like controls
The starting point in plugin creation is most likely the official documentation here: http://learn.jquery.com/plugins/basic-plugin-creation/

You can start from the very simple plugin that actually contain the function stick as follows:

$.fn.holder = function (options) { /* ... stick's code goes here ... */ return this; };


and this will be the base for your plugin.

Then later you can call

$('#control').holder({ ... some options ... });


And the defined method should be called.

This approach will probably change and below is the review of the code you actually posted, assuming that later you will change it anyway.

The coding style of the presented code is good. I only recommend to remove /***********variables*****************/ and /*************functions***************/ because they are usually become inconsistent with the code blocks below them. Even now the variables are mixed with expressions.

The logic here is unclear:

$('#textButton').click(function () {
    var options = {
        button: '#textButton',
        textBox: '#textBox',
        catcher: '#catcher'
    };
    stick($(options.textBox), options.catcher);
});


this in click handler is the clicked object - $(this) === $(options.textBox) so it it makes sense to move options outside of the click handler and use options.button in $('#textButton').

Also in strip it is not necessary to use $(textBox) because textBox is a jQuery object already.

It is not clear why you have added the functions addStrip and clearTextbox - they are used only once so their code can be just inlined.

$(textBox).val('');
$(textBox).focus();


The power of jQuery allows you to chain the methods (and textBox is already a jQuery object):

textBox.focus().val('');


And the main logic issue - currently the code on the halfway between pure jQuery implementation without model at all and model-driven approach (the code recreates model-like construction on every click). So it is better to select one of the ways and make a step:


remove holderArray and just use $(''.holder .strip'').filter(function () { return $(this).text() === text; }) instead of the array routines
or (and what is better) move holderArray and add code that keeps it consistent with the list. You can check my article on JavaScript MVC for more info on this approach: http://alexatnet.com/articles/model-view-controller-mvc-javascript