A living Pokédex is a fan term for a game which not only has a complete Pokédex, but has one of every available species of Pokémon stored in its PC boxes as well, usually in their National Pokédex order. It follows the most literal interpretation of the former slogan of the series, "Gotta catch 'em all!". It is considered a holy grail among some collectors. Among others, it is considered a waste of time and box space.
I'm working on my Living Dex in White right now, and I'm pretty close to finishing the first three gens. I have a ton of blank space after those since I don't have my Pearl version anymore, and didn't really catch a lot of the new Pokemon in White when I was playing through it. But now I'm catching up! I'll keep you guys updated with how it goes.
I plan on transferring all of them over to Bank/Y and starting a Kalos Living Dex, or at least of the Pokemon I don't already have AKA the new ones.