Header tags provide a way to make certain pieces of content stand out a little more in the view of search engines.
Search engines don't consider all text on a page to be equal. Paragraph or section headers within the page have more 'weight' than normal text on the page because they are considered to be more important.
How do you make a normal line of text become a header line?
Most content management systems (WordPress, Drupal, Joomla, etc.) offer the choice of a 'Style' for various parts of text within a page. You've probably seen references to "header 1" or "header 2" or "header 3" by now. Those are styles. Header 1 is considered the highest level style, while Header 2 is the next highest, etc.. Typically, a page might have two or three Header 3 levels, and maybe one Header 2.
The following three lines illustrate how respective header levels might display on a page, but every website can look different, depending on how it was designed.
Header Level 1
Header Level 2
Header Level 3
It doesn't matter to search engines how big the font is, or its color or such. All that matters is what header level it is. Words in Header Level 1, for example, are considered more important than words in Header Level 3, for example, and both are more important than normal, non-header text.