What is the quoting system? How does something get quoted? Is that the same as something getting moved? What is the purpose? I'm trying to catch on here....

I'm not exactly sure what you mean. Posts get moved when they are in the wrong location. For example if someone posted a recipe under the "Celiac Disease - Doctors" section, it would be moved to the "Gluten-Free Recipes" section. I'm don't know what you mean in terms of the quoting system though.
I'm not sure if this helps, but there are two ways to post a reply. If you click on smaller "Reply" link under a new post or reply, that post or reply is quoted in your reply. But if you click on the bigger "Add Reply" link at the bottom of the post, nothing is quoted. People use the quoting system in order to reference what they are replying to. Disregard this if it's not what you're asking, but I hope it helps.

-Brian

You can also quote from multiple posts within a thread, by clicking on the green "+QUOTE" button (which will then turn red and say "-QUOTE"--it's a toggle). Then when you use "Add Reply" all of the quotes will be included. You can delete parts of them, so that only the pertinent bits appear in your post.

By using this technique, you can even post a quote from a post in different thread (in a different forum even).

Quoting is useful when you are responding to a post several posts back in a long thread, as it establishes context. Overuse can be irritating, such as the person who makes the first reply to a long post, and feels the need to quote the whole original post.

Clicking on the little green right-pointing arrow the right of the data/time will take you to the post from which the quote was taken.