The answer changes with every generation.
When I saw the Nintendo Entertainment System in action I was awestruck. The Colecovision and both Atari systems had just become relics of a bygone era the moment I laid eyes on Super Mario Bros . Gone were the giant single shades of pixels and here was this little plumber with smooth animations and multiple abilities.
When the SNES was en route all I knew was things were going to be better, and boy were they. We got much more detailed sprites with even smoother animations plus deeper gameplay. While Sega had had a decent showing with the Master System, they now carved out a market all their own with the Genesis. Gaming grew up a little (at least into preteen territory) with attitude laden characters like Sonic the Hedgehog. Things got bloody with the arcade ports of Mortal Kombat II . The success of that game alone and the need to have it made Nintendo reverse their policies on gore.
The next generation changed things up entirely, it was all about 3D polygonal rendering. Whether you were being wowed by the adventures of Mario 64 and The Legend of Zelda: Ocarina of Time or delving into tombs with Lara Croft in Tomb Raider there was no going back to 2D sprites for the time being. It was at this stage that home systems really began to catch up to arcade machines. Dead or Alive, Virtua Fighter, Tekken, Bloody Roar, Tobal no 1, Battle Arena Toshinden ; the fighters kept on coming for Playstation. Console gaming grew up again, now taking on an additional amount of grit and realism.
N64, Playstation, and Saturn looked great but everyone saw room for improvement. By the time the Playstation 2 was all set to release people were talking mostly about graphics. I even recall some claims that it might be able to deliver Pixar quality visuals. That didn't quite happen but the leap was big, it was even pretty big right at launch. Gameplay continued to advance and controlling your three dimensional character was no longer a matter of turning them like a tank and pressing up to move them forward as was often the case. Intuitive button schemes became more common and the swivel camera was entering popular use. The first person shooter also began to take on its modern form.
People were worried a bit about the generation after PS2 superiority, would it just be a graphical upgrade? In truth that's about all it amounted to at first. It really took a few years before we began to see things that I think would have been capable of really surprising folks like the original advancement into 3D did. Ultimately this last generation has been defined by the conversion to high definition graphics, the prevalence of physics, and tons of freedom. Many things got better but those three really stand out. So the question becomes what can actually wow us in another age of high definition gaming?
The frontiers seem mostly explored at least until virtual reality becomes viable. Physics can get better, artificial Intelligence can improve, graphics can still get better, I suppose gameplay can be tweaked further; what can really be done to blow us away though? Some of the critics complaints about Knack and Killzone: Shadow Fall seem to circle around the notion that these games aren't next gen enough, whatever that means in a gaming climate like this.
I honestly have a hard time quantifying what would make a game appropriately shocking enough to be regarded in awe as a “true next gen experience.” You sort of know it when you see it. What do you need to see in PS4 games to tell you the next generation has truly arrived?