What is DOM element?

What is the meaning of DOM element in the following statements?

Statement #1

You can add multiple classes to a single DOM element.

Statement #2

The -- point is that inheritance in CSS is done not through classes, but through element hierarchies. So to model inherited traits you need to apply them to different levels of elements in the DOM.

I know that DOM stands for Domain Object Model. It seems to be synonym to API in the statements.

Document object model.
The DOM is the way Javascript sees its containing pages' data. It is an object that includes how the HTML/XHTML/XML is formatted, as well as the browser state.

A DOM element is something like a DIV, HTML, BODY element on a page. You can add classes to all of these using CSS, or interact with them using JS.

It's actually Document Object Model. HTML is used to build the DOM which is an in-memory representation of the page (while closely related to HTML, they are not exactly the same thing). Things like CSS and Javascript interact with the DOM.