Do I need a modded PS2 to download games to it?

May be...

I want to buy a PS2 and I'm wondering if it has to be hacked, just like the PSP to download games from the internet on a DVD and play them on the PS2.

Yeh, you need to get your PS2 chipped or modded to allow it to play downloaded games.

You should stay legal though - don't do it.