One <h1> per page... Headers are made for structuring the content on the page. The <h1> should sum up what the entire page is about (and should reinforce your targeted keyword phrase from your <title>). Since every page on your site should be targeting a sing keyword phrase (or possibly 2-3 if they are VERY similar), each page should have a single topic... and a single <h1>.
On your post page the <h1> is typically the title of the post.
On the home page post titles are typically <h2>s.
You don't HAVE to have an <h1> on your home page. But if you do it should target the same targeted keyword phrase(s) as the <title> of the home page (or a slight variation). It should generally reflect the overall theme of the entire site.
Multiple <h2>s and sometimes <h3>s are fine... But limit the number of <h1>s on the page to one!