Horses are amazing creatures. They need very little in the way of vocalization to make themselves understood. Almost anyone with a willingness to study the animals can learn how to read their body language.
Horse happiness is important because it is a sign of good health and contentment. Unhappy horses may be sick or injured. They should be approached with caution.
Look at the horse's head while it is in its box. A happy horse has a relaxed, alert expression. The eye of the horse is very revealing. An open, calm, friendly eye is a good sign. The nostrils and lips are also good indicators of a horse's mood. A sick or worried horse will have a "pinched" nostril or even signs of respiratory distress. The lips of a sick or worried horse are often tense and the face has a unhappy expression.
Next, look at the whole horse. Lame horses often have swollen legs. Injured horses may be bleeding. A horse that has torn off a shoe may not want to put weight on its foot.
Note any lethargy. Sick horses with fever are often lethargic. They may not want to move or interact with their surroundings. They may hold their heads down and fail to notice you when you approach.
Watch what the horse does. Sick horses with colic lie down and roll, walk the stall aimlessly, look at their sides, paw the floor, and show other signs of pain. This is a serious situation and a veterinarian should be consulted immediately. Horses that unhappy and bored in the stable (but not sick), may stall walk, kick the walls of the stable, paw the floor, and otherwise destroy their boxes. Always examine the horse's stall for signs of damage.
Look at the horse's feeder. It should not contain uneaten feed. Horses should eat their feed right away. A good appetite is a good sign that the horse is happy and healthy. If feed and/or hay goes uneaten, there is a problem. First, look at the feed to make sure it is ok (you should do this BEFORE you give it to your horse). Then, inspect the horse for signs of sickness.
Note the horse's interactions with others. A happy horse will often greet its people by coming to the front of the box with an alert, happy expression and maybe even a soft nicker. You can encourage this nice behavior by giving the horse a treat or a pat when you approach the box. An unhappy horse may not greet you when you approach its box, but may instead go to the back of the box and turn its tail to you. If this happens, open the door, but do not enter. Instead, call the horse to you with a treat. Repeat this a few times without riding the horse. It need to learn that its contact with you is pleasant and fun. A very unhappy horse may pin its ears, show its teeth and behave aggressively when you approach. This is serious and dangerous and you should get a trusted professional to help you.
Give your horse some companionship. Some horses hate to be separated from other horses. If a horse has "separation anxiety", it may be helpful to find a companion for it such as another horse, pony, goat, or even a barn cat. Horse toys, such as exercise balls, can also be helpful.
Try to turn the horse out as much as possible. Horses need fresh air, sunshine, exercise, and companionship to be happy.