All right, folks, here's what's going on:
The good news: it doesn't look like the database was affected, so once I get the boards uploaded once again, we shouldn't find any lost data or persistent problems with the boards unless there's something I'm missing.
The bad news: I still have no idea what really happened, though it looks like somebody just went into my web-hosting account and changed the HTML of all the index pages. I've changed my password, and will likely be doing so every so often for a little while, but I'm not sure how that'll stop hackers.
The other bad news: I've had to delete everything on my site to make sure that every trace of the hacking is gone. I'll be re-uploading the boards with the newest version of PHPBB, but it may take a while, and I'm afraid that it may not even be up tonight - I have work tomorrow, and I'm really tired.
I'll have another update in an hour or two.