They originally came out in the 90s in Japan, and were there called Pocket-Monsters (and still are, but in the "west" they are known as Pokemon).
They are a cardgame+videogame+cartoon+toys franchise, and is centered around the notion of a world where pokemon-trainers go out into the world to capture wild pokemon - you then use them to battle other pokemon trainers in leagues (so your now domesticated/enslaved pokemon form part of your stable of fighters).
There is the collecting aspect (got to catch them all - complete the set), and the exploring aspect (a recurring theme is that there are pokemon out there that have never been seen before, so think "brave explorers looking to expand the limits of human knowledge" kind of thing. The cartoon obsiously expanded on that with drama and more complex relationships between the characters/pokemon (they are sentient beings)
As it came out in the 90s, a lot of 30 somethings grew up with it, and are now presented with the idea that this thing which hit all their pleasure centers when they were young (adventure! excitement! challenge! responsibility! collecting! playing! etc!) is now real, and they can actually catch pokemon on their way to work. In fact, 25 to 35 somethings seem to actually be the largest demographic of players.
In case I made that sound like I dissprove of it, I certainly do not - I got a pikachu in a hatched egg the other day :) (In the new Pokemon Go game you find pokemon eggs, which you hatch by walking (the game records how far you have walked while trying to hatch eggs).
Oh, and in case you have missed the various articles floating about the internet, the new Pokemon Go game is an AR game (Agumented Reality - a game overlaid over the real world, very similar to geocaching) - it basically takes the "capture wild pokemon" concept, and overlays it onto a "google maps" like mapping system along with GPS tracking, so you have to walk around parks etc to try to find and catch pokemon, which you can then train up and then use to battle other pokemon trainers in "Gyms" (which are locations in the real world which if you are close to physically, you can visit in the game itself)