Using AJAX and WordPress

All,
Currently I have some JS in an external JS file called external_js.js.

In here I have the following code:

$("#submit_name").click(function() {
category = $("#category").val();
search_term = $("#search_term").val();

$.post("http://localhost/wordpress/wp-content/search.php", { category: category, search_term: search_term }, function(results){
    $("#search_results").html(results);
});
});

This is basically an AJAX request to this file:

$themePath_config = ABSPATH;
set_include_path($themePath_config);
include "wp-config.php";
$category = mysql_real_escape_string($_POST['category']);
$search_term = mysql_real_escape_string($_POST['search_term']);

$qry = "Select wp_posts.ID, wp_posts.post_name, wp_posts.post_title,
from wp_posts
join wp_term_relationships on wp_posts.ID=wp_term_relationships.object_id
where wp_posts.post_title like '%$search_term%' and wp_term_relationships.term_taxonomy_id='$category'";
$result = mysql_query($qry);
$resultrows = mysql_num_rows($result);
echo $resultrows;

Here is my page inside of WordPress that I’m trying to call my jQuery/AJAX with:

[tabs]
[tab title="I choose as a vendor..."]
<div id="choose_past_vendor">
<form id="past_vendor_form" name="past_vendor_form" method="post">
As my vendor for my
<?php
$qry = "Select wp_terms.name, wp_terms.term_id
from wp_terms
Join wp_term_taxonomy on wp_term_taxonomy.term_id=wp_terms.term_id
where wp_term_taxonomy.taxonomy='category'
order by wp_terms.name ASC";
$result = mysql_query($qry);
?>

<select name="past_vendor_category" id="past_vendor_category" data-placeholder="Choose Vendor Type" class="chzn-select">
<option value=""></option>
<?php
while($resultset = mysql_fetch_array($result)){
echo '<option value="'.$resultset['term_id'].'">'.$resultset['name'].'</option>';
}
?>
</select> I choose: <span id="past_vendor_selected_name"></span><br><br>
Enter in the vendor's name below: <input type="text" name="past_vendor_name" id="past_vendor_name"> [button id="submit_past_vendor_name" style="gray"]Find Vendor[/button]
</form>
<div id="past_vendor_search_results"></div>
</div>
[/tab]
[tab title="I recommend as additional vendors..."]
Tab Content
[/tab]
[/tabs]
<script type="text/javascript">
  $(".chzn-select").chosen();
  $(".chzn-select-deselect").chosen({allow_single_deselect:true});
</script>

I’ve read some articles that say this should be moved to admin-ajax.php but I’m not sure how to go about making these changes.

I was trying to use this article but I’m just not sure how to implement this. My button for the submit_name button is on a page that I created from within WordPress. Here is the article:

http://www.garyc40.com/2010/03/5-tips-for-using-ajax-in-wordpress/

Can anyone point me in the right direction?

Thanks in advance!