I'm trying to find a tutorial or script that has most of the Google Instant functionality but I would be hit my API instead of Google and then displaying results from my domain.

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Here is a tutorial: http://woorkup.com/2010/09/13/how-to-create-your-own-instant-search/

You can also look at the source of the many available jQuery plugins for that purpose (google for "live search", "instant search" or similar). For instance, this one: http://ejohn.org/blog/jquery-livesearch/

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It's just an AJAX autocomplete. jQuery UI has a built-in autocompleter that's very easy to work with.