I want to make a static page for popular posts from three different categories. Are there codes that I can put on the page that will display all the posts from each category under a header or should I type it out myself.

There is no way to automatically display three categories on a static page.

You can create a custom menu and include the autmomatically generated Categories pages the software provides when we publish a post.
http://en.support.wordpress.com/menus/
http://en.support.wordpress.com/menus/#adding-category-pages-to-your-menu
http://en.support.wordpress.com/menus/#changing-the-order-of-menu-items-and-creating-sub-menus

I want to make a static page for popular posts from three different categories.

Are there codes that I can put on the page that will display all the posts from each category under a header or should I type it out myself. I know there is an archive code and it lists all the post I have created but was wondering if there was a way to do it differently.

Thanks for your help!

Arianna