If my company is called foo.com and I am on the homepage/index page of the site, what should the <h1> and <h2> tabs be?
<h1>foo</h1>
<h2>home</h2>
If I am on the settings page for example, should I use the following?
<h1>foo</h1>
<h2>settings</h2>
Or should I be using the <h1> and <h2> tags for something else?
Google and other search engines usually read things like recommendations from the W3C and RFCs. According to a W3C recommendation for The global structure of an HTML document:
A heading [h1,etc] element briefly describes the topic of the section it introduces. Heading information may be used by user agents, for example, to construct a table of contents for a document automatically.
In SEO we strive to help the search engine better understand our code, layouts, etc. For the best SEO, you should be using rich snippets and checking them with the rich snippets testing tool. If you show the search engines you're serious about organizing your content, and they serve people your pages and they stay for said content, then you will be rewarded.
I have to believe there has to be a more detailed thread regarding this StackExchange. Please dig a little deeper. Also you need to determine what you are trying to accomplish with your Heading tags. Whether your motives are related to SEO or accessibility that should be where you start and go from there. Please see this page from Penn State University regarding the subject.
The heading tags should be used to show the importance of the content that they represent. The tag is therefore the main heading for the page, and the should represent the least important heading. The should therefore show what that page is about, and so should be used for 'settings' in your example.
It is important to note that it all depends on your website though. if you have an article (blog) website you may choose to use the site name. In essence the should reflect what the page is about and where possible shouldn't be duplicated across the site as this may be regarded as keyword stuffing and duplicate content.
I would recommend simply placing the site name in a div, or an h tag with lower promenance ( for instance) and styling accordingly with CSS
