If you are looking to have a unique heading (i.e. H1 style) on only your home page, you can do this two ways.
The preferred method is to create a custom style for just the H1 on your home page. Most themes generate body classes, such as 'home', 'page', 'blog', etc. Using the body class of your 'home' page you would add the following to your style.css file in the theme directory:
body.home H1 {
(your custom style here)
}
Another method is to modify your template file with a custom query and insert your own code for just the page/content you specify. If you are changing the H1 of just blog posts on your homepage, you would modify Twenty Twelve's content.php file with something like this:
<?php if (is_home()): ?>
(your H1 line goes here)
<?php endif; ?>
You'll see similar code in the header section in content.php which will show you how the template handles whether the page is single post or not. You can tweak this pretty much any way you want. A list of body classes can be found here http://codex.wordpress.org/Function_Reference/body_class
It is ideal to keep your presentation changes (style) separate form your structure (layout). Using CSS approach instead of changing your template is recommended if you only want to change the style of the H1. If you want a custom layout you might want to explore creating a custom template from scratch. More on this approach can be found here http://codex.wordpress.org/Pages#Creating_Your_Own_Page_Templates
I hope this helps!