Heading tags, specifically <h1>, can be used to denote important content on your site, however just like with your title tag you must think in percentages.
For instance you could use CSS to make your H1 tag look like normal text and then use it for each paragraph of content. However, that would be less beneficial than just putting a few choice words in H1 tags as when you put an entire paragraph in them the keyword density for any one word is going to abysmally low.
So, do not try to do anything tricky like that, the search engines well ahead of you on that point and it will not work. Instead use the H tags as they were meant to be used.
Use H1 for the on-page title (such as the article title at the top of this page). Use H2 for secondary content sections, H3 for tertiary ones, and so on down the line just as you would if you were making an outline.
You can of course use CSS to change the appearance of your tags, to make the fonts smaller or larger or a different color, this will not affect you rankings. The content doesn't have to be a big font, just appropriately used as a header and with good keyword density.
In years past search engines also gave higher weight to content found in <b> or <strong> tags or other tags that indicated emphasis. It is hard to say if they still do, since it is nearly impossible to scientifically test, however even if they still do this any such bonus would be small and not worth worrying about. So again, use the tags as they were intended and do not think to try to eke out a benefit by overuse or through manipulating them to appear like normal text when they're not.