The question did not specify a DB product or programming language. However it can be done with ANSI SQL in a cross-product manner.
Assuming a programming language that uses $var$ for variable insertion on strings.
On the server you get all selected values in a list, so if the first two boxes are selected you would have a GET/POST variable like
http://url?colors=black,white
so you build a query like this (pseudocode)
colors = POST['colors'];
colors_list = replace(colors, ',', "','"); // separate colors with single-quotes
sql = "WHERE ('$colors$' == '') OR (color IN ('$colors_list$'));";
and your DB will see:
WHERE ('black,white' == '') OR (color IN ('black','white')); -- some selections
WHERE ('' == '') OR (color IN ('')); -- nothing selected (matches all rows)
Which is a valid SQL query. The first condition matches any row when nothing is selected, otherwise the right side of the OR statement will match any row that is one of the colors. This query scales to an unlimited number of options without modification. The brackets around each clause are optional as well but I use them for clarity.
Naturally you will need to protect the string from SQL injection using parameters or escaping as you see fit. Otherwise a malicious value for colors will allow your DB to be attacked.