Can anyone recommend a good source for a FREE PHP script to handle
tracking of links to my site?  I would like to place ads and targetted
links throughout the Internet and track those customers that return to
my site and eventually signup by clicking through those links.

If your affiliates use a link format like;
<a href="http://www.yoursite.com/ad-track.php?adcode=214310342405">Your Link</a>

and you have "adtrack.php" on your server with the following bit of PHP in it;

<?php
$fp = fopen("ads.txt", "a");
fwrite($fp, $adcode);
fclose($fp);
?>

then "ads.txt" will be a flat file database of your ad campaign. Just
put a different adcode for each place you put up an advert. You could
sort the data in "Exel" or whatever and see who is delivering more
hits to your site.

If you want to track them through your site you will need access to
your log files. Providing your server logs retain the "referer" tag
then your adcode variable will show up next to the IP address of the
visitor. Simply examining which pages that IP address visited should
give you a good idea of where they went. You could also try these;

http://www.scriptdungeon.com/free_php_scripts/log_analyzers/free_php_scripts.php

Hope that helps.