>As I understand ,the search engines that don't use Meta Tags create the description (that appears on their listing) by taking the first few sentences of your content. However, if you have an H tag that precedes your first paragraph, does the text in your Heading tag get included, or does the engine skip the H tag and only include the sentences starting within the paragraph?
Yes, the text inside Hx tags is usually included, as would any other text. That explains the bizarre "descriptions".
To prevent this happening to your pages, make sure that the FIRST bits of text are REALLT what the page is about and not links to elsewhere. To decide what are the first bits, look at the source code ... read it straight through and see what comes first. Often is it a table-based layout, and the list of links will be used for the description. Change the layout so that you lead with the H1 and a solid opening paragraph.
HUMANS read those descriptions, and will skip through the returned search results looking for a description that looks like what they need.