I have actually create a similar search application. The best way to accomplish this would be to use an AJAX script to add variables, based on selection criteria from the check boxes, to the URL, such as this: www.mypage.com?computer=dell&othervar=anothervalue&nextvar=athirdvalue
Then, just as Eric stated, you use recordsets where you would filter the recordset based on the appropraite column containing the data in the URL variables.
such as: SELECT * FROM table WHERE computer = '$_GET['computer'] AND othervarcolumn = '$_GET['otherval']
You may need several record sets to get it done to exact spec as you want. And you may also need a lot of "if" statements.