AR Cards are cards that come with the Nintendo 3DS console. The word "AR" stand for "Augmented Reality". It makes use of the 3DS Camera. Scanning the card with the camera will make something pop up onto the card on screen, depending on which card you use. For example: using a card with Mario's picture on the front will make a figure of Mario will pop up onto the card.
Nintendogs + Cats
Nintendogs + Cats makes some use of this feature. There is an item called the AR Camera which works like the 3DS Camera, scanning the card except this time, your own pet will appear on the card. However, if for example you use a card with Link on it, your dog or cat will appear on the card wearing a Link hat, and if you use a card with Pikmin on it, your pet will wear a Pikmin hat etc.
The Obedience Trial contest utilizes an AR Card which, as said before, will make your pet appear on the card. You need to keep the camera focused on the card and use your commands to tell your pet what to do.
3DS: "This curious camera lets your pets jump into the real world!"