Although I recomend using an external link to the CSS as soon as you learn how.
The <H1> tag needs to be optimized in much the same way as your title tag. They should always be used together. You should use your H1 tag on your first sentence or phrase in the text of you page. The H1 tag should target your main keyword phrase and be a brief description of the text to follow. Think of it as an outline, your main point should be in an H1 tag and then you should write about that point. Your next main point should be in an H2 tag, all the way down to H6. You should not put H1 through H6 on your page just for the sake of having them. If you have 3 main points use H1, H2, and H3. Every major search engine uses these tags. They are a way for you to tell the search engines how to organize your page, and what you think is important. Do not abuse them, if you use them like an outline format and place relevant keyword phrases inside of them you will really help your web page achieve higher rankings.
One down side to the <H1> tag is that the font inside of it will be very large. This is why you should use CSS to control the look of the text on your site and inside your tags. If you do not know how to write CSS yet just paste this code between your Head tags on your page.
You can change the colors, font, and size to match your site. If you you need the H2 tag and so forth just duplicate the H1 part and rename it H2 all the way to H6.