Next Generation Console
The Term Next Generation Console (or Next Gen Console) was first used for the PlayStation 2. Reason for that being that it was the next generation after the PlayStation. This term has been used for the next "next generation" - following the "first" next generation - as well, even though they are not from the same generation.
With time the term "Next Generation Console" has been applied to all the rumoured or planned incarnations of then current consoles; e.g. when the PS2 was current, the PS3 was referred to as "next gen", but when the PS3 was current, the PS4 was "next gen", and so on. So, at different times in the past the term "next generation" referred to different sets of Video Game Consoles.