We didn't get the opportunity to check out the PlayStation Vita's built-in augmented reality games when we received our Japanese unit -- because the Japanese unit doesn't come with this stuff. But you can see the "Table Football"/"Table Soccer" game being prepared for the Western launches in this video from the US PlayStation Blog.
Table Soccer (we're American) allows you to turn the surface of your choice into a playfield by laying out AR cards. You can adjust the size of the area by spreading the cards out, placing the bleachers and scoreboard with specific cards. Then you play soccer by flicking the ball with your finger. It's like all those table games you played in elementary school, except way more expensive and futuristic, basically.
This is one of three AR games built in; the other two are Cliff Diving and Fireworks.