I always hear people talk about DOM this, manipulate the DOM, change the DOM, traverse the DOM; but what exactly does this mean?
What is the DOM and why would I want
It means working with the Document Object Model, which is an API to work with XML like documents.
From w3 on the DOM:
The Document Object Model is a platform- and language-neutral interface that will allow programs and scripts to dynamically access and update the content, structure and style of documents. The document can be further processed and the results of that processing can be incorporated back into the presented page. This is an overview of DOM-related materials here at W3C and around the web.
One of the functions mostly used in DOM work is:
getElementById
Manipulating/Changing the DOM means using this API to change the document (add elements, remove elements, move elements around etc...).
Traversing the DOM means navigating it - selecting specific elements, iterating over groups of elements etc...