You earn this achievement by playing
(EDIT: According to mcmanus1986
: A patch seems to have made this: Play 25 UNO games
) full games of Uno. You don't need to play
25 games in a single session; this one's cumulative.
Make sure that you let the game explain to you how Uno works in the Tutorial, for that game counts as a played game as well.
Missed that opportunity? Don't worry, you'll get this achievement naturally by just playing the game. I got this one when I was a level 5 player, so that is quite early in the game, considering you have to grind to level 100 for all of the game's achievements.
Want to know how many games you played already? Just tap 'Achievements' on the Main Menu. Here it tracks all of your achievement progress, including this one.