If the PSone put Sony on the game industry map, the PlayStation 2 helped them conquer it. Selling over 100 million systems worldwide, it's rivaled only by the original Nintendo Game Boy and newer DS in terms of widespread popularity and prolific hardware sales.
Originally launched in October of 2000 (I was in line with everyone else), the system faced limited supplies that fell way short of the huge demand, fueled by several key features, chief among them being hardware backward compatibility to PSone games and built-in DVD playback functionality.
After besting initial development hurdles, the PS2 saw a flood of original content (Jak and Daxter, God of War, Ratchet and Clank, Persona series, etc.), cementing it as a favorite for the ages worldwide. People used to see "games" and "Nintendo" as synonymous. Anymore, it's just as common to hear someone say they want to play "PlayStation" as it is to hear "Nintendo."
The little black box that could continues to be popular due to price and size reduction and widespread ownership, as well as Sony's questionable efforts to remove PS2 game hardware backward compatibility from the PlayStation 3.