I thought it might be fun to share what everyone considers to be games that most influenced their current tastes whether it's something they played when they were younger, or something that had a profound effect when you got older. So I'll start first:
My first gaming experience where I was fully cognizant of what was going on was probably our Sega Genesis. We didn't have too many games for it, but one of the games that most impacted me was Streets of Rage 2. I remember spending days and days with my brothers trying out different characters, bowling through strangely named punks and brawlers, and all the while fighting to the EDM stylings of Yuzo Koshiro. It was an extremely stylish and solid game that embedded in me a love for brawlers 2D or otherwise.
I also have a fondness for RPGs and I attribute a lot of that to my stepfather being a big Dungeons and Dragons nerd. He would try and get little games set up for us when we were probably too young to fully understand what we were doing, but there was an allure to the freedom the game provided. We were building people out of nothing but dice rolls and character sheets, using our imaginations to decide what our character was like, and we got to make the decisions on what we wanted to do in the world. It was fascinating! Outside of D&D I would watch my stepfather play Final Fantasy late at night on a borrowed NES. It was a similar experience in a lot of ways. It was something I didn't fully understand but it had characters you had control over, a wide, mysterious world with little direction and the most common encounters were from strange, hostile monsters. It's a feeling I hearken back to when I encounter games with the same kind of atmosphere to it. It's also an atmosphere a lot of PC RPGs are home to, so I tend to stick there.
Phew! I hope that wasn't too personal :X, but I'd like to see what everyone else's experiences are.