For the emerald, amethyst and sapphire medal, it means games. I finally got the amethyst medal yesterday after way too many hours. If you don't know already you can see where you are at under the statistics, so once you clear 1250 you get the emerald.
Now for another 1250 puzzles to get the sapphire!
I infer that what you're saying is the "Tranquility" achievement comes after completing four games. (25 puzzles per game x 4 = 100 puzzles)
A game consists of completing 25 puzzles, at least logically, since after you have done so, it ends. You get offered to "play again" but it is not like in Origin where you continue playing the same game and accumulating score. Presumably the way it's worded
you could complete a single puzzle in Relaxation, quit, restart a new game and the single completed puzzle would count toward the 100. I'll experiment to see if it increments the puzzle counter that way.
The awards / achievements terminology is confusing. If a "set" is a completed game, then why not just say complete 100 games in Relaxation mode.