Correctly Printing $search_box and adding Style

Hello all,

I am new to theme dev on Drupal. I have succesfully added a searchbox and positioned it on my theme appropriately using this call in page.tpl

<div id="right-search">
               	<?php if($search_box): ?>
				<?php echo $search_box; ?>
				<?php endif; ?>
                </div> <!--/#right-search-->

I'm confused by the output of:

<div id="edit-search-theme-form-1-wrapper" class="form-item">
 <label for="edit-search-theme-form-1">Search this site: </label>
 <input type="text" class="form-text" title="Enter the terms you wish to search for." value="" size="15" id="edit-search-theme-form-1" name="search_theme_form" maxlength="128"/>
</div>

I'm not sure where its getting that ID "edit-search-theme-form-1-wrapper" as well as the other labels and ids. Is there a way for me to add a custom ID and style it appropriately?

Thanks for any help!