There are four main techniques to integrate the Search Appliance with your site. The techniques are grouped as follows:
The first two are simple to implement. They involve only static HTML content, and can be used in situations where no dynamic scripting is available.
The latter two are more powerful and can be used in dynamically generated web sites. You'll create a "search page" for your site (search.php, search.aspx, etc) which accepts a query from the search user. Your search page makes a search query (HTTP request) to the Search Appliance, and receives result data. Your search page then displays the data in whatever manner you wish to the search user.
The advantage of the latter two methods is your site controls all interaction with the search user, making it easy for your search results page to "inherit" the look and feel of your site. They never contact the Search Appliance directly.