How did Dani get the "Search" link to go directly to the advanced search page?

Also does the Search field (see the purple-colored bar near the top of the page) perform the same search as on the search.php page?

I'm a bit confused as to what you're asking. The word "Search" is a direct link to the search.php page. I used CSS to make it not really look like a link, but it's just a standard hyperlink. The search field is in a simple form which calls the search.php page. If you are running a vBulletin board, you could put the following wherever you want:

<form action="search.php" method="post">
	 <input type="hidden" name="do" value="process" />
	 <input type="hidden" name="showposts" value="0" />
	 <input type="text" name="query" />
	 <input type="submit" value="Go" />
</form>

Then, just play around with CSS to make it look the way you want :) Notice that when you do a search, a list of threads is returned. If you change the showposts attribute to 1, a list of individual posts will be returned. Your preference :)

Also, since it's just a regular ol' form - that HTML code can go anywhere ... within a vBulletin template or on a regular HTML page.