Where is the code for the search box?

I am looking for the code that generates the search box, is that part of the search module? If I wanted to overwrite, would I have to add it to the template.php file within my new theme? I am confused as to how some of these variables are actually constructed and I am seeing the template.engine file but I don's see it there. Any explanations to get me out of the darkness would be really appreciated.