customize search form widget wordpress

Check out this page. When the search form is requested via PHP, WordPress checks the active theme for a searchform.php file and displays the contents of it. Failing that, it uses it's built-in form:

<form role="search" method="get" id="searchform" action="<?php echo home_url( '/' ); ?>">
    <div><label class="screen-reader-text" for="s">Search for:</label>
        <input type="text" value="" placeholder="Search Site" name="s" id="s" />
        <input type="submit" id="searchsubmit" value="Search" />
    </div>
</form>