So what happens if tons of people go to your site, and it goes over the amount of bandwidth for your plan? does your whole site shut down or what?

here is the kbase link on subject, it will answer your question:
Click here

Or perhaps this link which is more relevant to Dreamhost customers (as opposed to dedicated server customers).

And the overall Dreamhost >> Bandwidth section:

oops, didn’t realize i had selected “dreamservers”, my bad.