How can I get my Google Search Bar on my site to display my site's results?

It has nothing to do with google. Search results are compiled from meta tags. All website/pages that show up in searches (google, yahoo,msn, etc.) are found and sorted by their meta tags, which are embedded in the pages html.

Go to any web page, click the view menu, page source. You'll see something like this at the top: (this page for ex.)
<meta http-equiv="Imagetoolbar" content="">
<meta name="description" content="I recently added the google search bar to my site,...">
<meta name="keywords" content="yahoo, answers, questions, Google">