Reasonings:

Atari 2600 - Made home consoles popular. Without this you've got clunky 'home arcade' machines with primitive graphics..a console, two joysticks and a TV are templates that haven't changed 30+ years later.

NES - Outside of the obvious (game sales and debut of various series that still exist today) it's the console where game development took a leap forward...your JRPG, sports game, action game were all beginning to get perfected on this system

N64 - The death knell of the sprite era and the beginning of the forever reign of 3D. N64's 3D offering was the first time you saw games not just re-skin games that couldda been mainly sprites into 3d but actually take advantage of the space (Mario 64)

PS1 - The beginning of 'adult' gaming and having limitless storage space due to cheap CD's helped another push forward in game design

PS2 - While Atari made home consoles popular, PS2's made them mainstream. The first 'adult' gaming system and technically the first to introduce both hard drives and online gaming into the console space.

Xbox360 - Seemless online/offline video gaming, mandatory storage space, basically mandatory online... probably the easiest game development pipeline ever...its the template for modern gaming in this generation and barring a crazy leap in tech (RIP VR) for many more.

Honorable Mentions:

OG Xbox - disappeared the barriers between PC game and Console game development and pushed forward online gaming in the console space.

Sega Genesis - While SNES was just refining their NES classics, Genesis was making games more adult (Mortal Kombat), and perfecting the biggest video game genre in the world, sports games.

Turbographics 16 - First enthusiast system. It felt like previous console releases focused on wowing tech geeks and rich soccer moms kids for those initial overpriced sales. It was the first time a console was marketed towards the hardcore, for better and for worse.

Poll question: Most Influential video game console of all time?

Poll result (28 votes) Atari 2600 - The Don Dadda of this shit (1 votes) Vote NES - 90's Chicago Bulls (17 votes) Vote N64 - First 3D only system...the death of sprites (2 votes) Vote PS1 - Leap to CD tech can't be underestimated. Games could grow unrestrained (4 votes) Vote PS2 - So popular we can even argue it killed off VHS (3 votes) Vote XB360 - Modern gaming defined. (1 votes) Vote