Hi. I have a six year old daughter. She has previously been all ages below 6. Her friends and assorted neighbor kids who show up at my house range up to age 13.
I also happen to have a 15 game basement arcade, most of which play multiple games.
These age ranges come from my personal experiences. I have found that while their are a few prodigies, children who aren't on my schedule are actually more likely to be behind their age than in front of it. I blame modern consoles being impossibly complex for young kids and tablets having controls that don't transfer to real life.
Age 3-5. They basically can't actually play anything. Every once in a while you will get a prodigy, but the average kid in this age range can't clear a single screen or level, most of them will struggle to even make the connection between the controls and the actions onscreen.
Age 6-7. They can start to play some games. Driving games and single button games are good. My daughter prefers slow speed boy Pac-Man as her favorite game. Kids this age will finish the first level or so of many classics.
Age 8. Puzzle games become playable, they can handle beat em ups.
Age 9+. They can play anything.