Signs of happiness in dogs are very similar to signs of affection. When your dog is happy, it will most probably show the following behaviour.

Photo by Diane
• Playfulness: bouncing, jumping, circling or running back and forth.1
• Being friendly to the touch.
• Curiosity. Happy dogs like to explore the surroundings.
• Friendly barking.
• Relaxed body and posture.
• Moderate excitement.2
• Perky and pointing forward ears, or ears slightly back.
• Relaxed and slightly open mouth.
• Relaxed brows and eyes.
• Relaxed neck without any signs of stiffness.
• "Smile" on the face.3
• A tail at neutral height (about the same height as the rump), wagging in a graceful, pendulous , smooth and relaxed way. It may wag so hard that the entire body wiggles back and forth with the movement.
• The hair laying flat.
• Eagerness to obey your commands.
• Rolling over.
• Normal stool and appetite.

Please note that any of these signs taken separately can also mean other things including dominance. To understand your dog right, you should always pay attention to its body language as a whole, not just a single sign. Besides, every pet is different. Knowing your pet is the best way to keep it happy.
1 Unless they are obsessive.
2 Some dog behaviourists believe excitement is never a sing of happiness. They are sure dogs who act excitedly do so because of a lack of exercise, discipline, and consistent training from the owner. From this point of view, excitement becomes a sign the dog is not mentally stable enough.
3 Make sure you and your kids don't confuse this friendly smile with an aggressive grin.

See also: Children Think Angry Dogs Are Smiling and 20 Utterly Happy Dogs