You'll be training your pets by Praising/Scolding and making the pets learn 20 traits. Should be collective, because I believe in there's only 11 unique traits the game gives you. Also it helps getting more than just one pet (which probably lends to the achievement's name) since only one pet can have 5 traits taught.
One warning: it seems this trait was bugged. When I tried playing it, it didn't really matter how many traits I did teach, I could have 25/20 and the achievement won't unlocked. It's been a few months, but I haven't heard any update on it.