Describe the purpose of various elements like: <header>, <footer>, <nav>, <section>, <article>, etc.

Header could be a name or logo. Nav is short for navigation which has information for links and secondary files. The footer usually has information that tells us about the author and has all copyright data and all information sourced to generate the document. An article is an independent item of content that exists within the HTML. The purpose of these elements is to help to create and format the content to allow the flow to be natural. It is general structure.

How are they different from and similar to the <div> element?

A div is a block level element used as a container to hold contents together. It basically acts as a boarder to keep specific content grouped together and still maintain the flow of the data. This is a form of customization that allows stylization to enhance the visual aspect of the content as opposed to the other elements which focus more on structural format. The div is a diversion, or variation, of that information. This does not include CSS customization of the color, font, size etc. It is a grouping tool.