Your WordPress installation comes with an RSS Feed. From any other website you can tap into this feed to extract the latest information of that Blog. I do believe that this method has a 12 or 24 hour delay. So the RSS Feed in this method is only updated once or twice per day. Meaning if you post a new post on your WordPress site, it can take up to 12 hours before the latest item is added to your footer on the other site.
On your other site, add this code to your footer
<?php
$url = "http://example.com/rss/feed/url/goes/here";
require_once(ABSPATH . WPINC . '/rss-functions.php');
$rss = fetch_rss($url);
?>
<ul class="footer-links">
<?php
foreach ($rss->items as $item ) {
echo "<li><a href='".$item['link']."' title='".$item['title']".'>".$item['title']."</a></li>";
}
?>
</ul>
Be default, your RSS Feed URL is simply your Blog's base URL, followed by /feed
You could alter the foreach loop to limit the number of links presented.
The easiest way, though, is to go to your WordPress Settings > Reading and set the value for Syndication feeds show the most recent. Default is 10 posts.