This is another topic that has been discussed for hundred of times, and yet, people still ask if Header tags (h1 h2 h3 h4) are important for search engine optimization.

Lets see what is the use of Header tags and why they are necessary for your site.
The header tags are VERY important for SE’s (Search Engine’s), they help SE’s to determine and understand what the page is about, so it can give a certain importance to that page while a user searches for determinate keywords.

Search engine algorithms are designed to take advantage of properly built sites. This includes the use of the heading tags to denote headline or other important text on a web page. Most engines give some boost to text found within the headline tags of a page. For this reason, it is essential to make use of the < h1 > tags on optimized pages.

The header tags should be used as Title of content; it should be nice and be able to give a relevance of the content, what the content is about. ( Don’t Confuse it with the title tag, header tags are within the body of the page)

Like in this page the title is "Make Money Online - Header tags relevance" and Header tag is "The Importance of Header Tags for Page Rankings"

What are Header Tags?

Header tags label your headline so that search engines will recognize it as "important" on your web page.  The header tags will make your headline bolder so it will stand out.  The following examples explain further

<h1>Make Money Online</h1>
<h2> Earn Living Online</h2>
<h2> Use Email to Bring Visitors</h2>
<h3>How to Send Emails</h3>
<h3>How to Write an E-zine</h3> 

What are Different Types of Heading Tags?

Notice in the example above there are three different heading tags (h1,h2,h3).  These indicate not only the size of the heading but also the importance.  Once again, the main headline will need an "h1" code to give it the greatest importance.  The main sub-headings, which are numbers one, two, and so deserves a "h2" showing they are next in line when it comes to importance.  The "h3" can be given to sub-headings under the main sub-headings.

Note do not overuse these tags, but applying the < h1 > tag to the most important text on your page and smaller header tags like < h2 > and < h3 > to section headers, caption text, etc... is good practice. If the standard font size denoted by the header tags is too large for your design, you can make use of style sheets to redefine.

When the importance of a heading is established with HTML tags, you're actually making it easy for the search engine robots / spiders to determine what your web page is all about.  The search engine doesn't have to guess at it.  It can see exactly what your main topic is and place you under the appropriate keywords.

How to Write Your Heading

Body Content

Page text is very important part of search engine optimization. Many search engines like Google "strip" the body text and use it to not only to display sections in their search results, but they also use the body text to determine the subject matter of web pages.
Keywords and phrases should appear toward the tops of your web page and "trickle" down to the page.

Here are example Heading tags in order which should appear in-between the BODY tags of your web page(s):

<H1> 1st Business Heading</H1>
<H2> 2nd Business Heading</H2>
<H3> 3rd Business Heading</H3>

When writing your headings, include your main keywords and key phrases in the headings.  Be as specific as possible to increase your chances of being listed in a reasonable spot in the search engines.  General search terms or keywords usually generate poor indexing because so many websites use them. 

Examples:

General keyword:  Make Money
Specific keyword:  Make Money Online

Headline:  Guide to Making Money Online for Beginners – Learn How to Earn

Also, don't forget to optimize your entire web page, not just your headings.  Your headings are only a small part of SEO. 

I believe that SE's like Google view the text inside heading tags as important. After all, it does set the tone of the entire web page. The same importance is applied to text that is BOLD and/or in ITALICS. But DON'T over do it. If a page text is mostly headings, in bold, and/or in italics, it may not appear as important. It certainly wouldn't to the viewer.

The paragraph(s) to follow would then be a summary of the heading. Somewhere inside this summary should be your keyword phrase. In my opinion, use NO MORE than 1 keyword or phrase for every 100 words.