Get AJAX to send a request to a php script when a specified event occurs.
The php script then retrieves the required records from the database using mysql_query.
Loop through the returned records in the php script and generate the html for the list - a <select>, <ul>, <ol> or some other list (you didn't specify the type of list)
The AJAX function then receives the html returned by the php script.
Output the returned html to your web page.