Interestingly enough, it was the original Super Mario Bros. for the NES. I was 2 years old (almost 3) when I played it. And I have been a Nintendo fan ever since. 
yesIt may not have been your first game ever, but what game made you realize how much you loved to play them or made you actively competitive?
The game that made competitive first was Mario Kart on the Wii. The first game to make online competitive was Call of Duty: Black Ops.