DDoS stands for “Distributed Denial of Service”. Denial of Service means you occupy certain resources for a long period of time, making other people not able to use them. A real world example would be running up to the cashier at a super market and paying in small coins, counting every single one really slowly, then finding out at the end you didn’t have enough in the first place.
The “Distributed” means it’s done from a lot of places at the same time, so it’s much harder to simply deny access to the clients that do that. Usually this is done via computers infected with trojans, so people participate in this without even knowing they’re doing it.
As to why minecraft.net is being targeted, I’m not sure.
While the guys hosting my server are looking into things, I’ll get back to coding.