And since the attacks were done so close together in time frame, surely it had to be the same person, wouldn't you think?
Not at all. Port scans and simple attacks are quite common.

I've read it has something to do with attacking DOS, which I think is the basics for any computer isn't it?
Erm...... well, sort of. But not really. It's kind of hard to explain in a short post.

Basically, DOS is the foundation of Windows 9x and Me. If you're using Windows 2000 or XP, you have no DOS in your system, so you have nothing to fear from that attack.

BTW, I know nothing about the particular attack in question. I'm just speaking generally. I recommend a good Google search for your attack to find out more.