The reason is kind of complicated, but I'll try to explain it. Forums work because they're hooked up to databases. From a technical standpoint, the database is a vitally important facet of the forum. It's where all the posts, member information, topics, private messages, etc, are stored.
We have our website hosted at Point A.
We had our forum and database hosted at Point B, a place totally unrelated to Point A.
Our website at Point A uses something called a Content Management System, which is hooked up to the same database at Point B (you don't need a database for a website, but we use one). Apparently, the hosts at Point B won't let us hook our database up to an outside website – Point A.
Therefore, the website went down because it could no longer connect to the database. Since the hosts at Point B wouldn't allow us to do what we wanted, we picked up our database and put it somewhere else that would (another web hosting service).
When the forum was down, we were moving our database, doing a reinstall of the forum software, and repairing a few things.