Can't find my website page listed under admin/structure/pages
The /contact page is provided by the Contact module, which you must have enabled. It displays the contact_site_form form, which is declared in /modules/contact/contact.pages.inc.  

Configuration options for this form can be found at the site.com/admin/structure/contact URL.

If you're looking to change form structure, you may need to create a custom module and implement hook_form_alter(). If you're after changing layout, you may be able to get away with adding/moving blocks (in which case have a look at Context for more options than the core blocks page), but more likely should use something like Panels to create a new page and insert a new contact block on it. You may want to have a look at the Webform module, which can be used to create &amp; configure forms through the UI.