Horses are herd animals, they desire to be with other members of their kind.
If not separated from their herd on a regular basis some horses become herd bound and suffer from anxiety when taken from their herd or when another horse is taken.
The problem is best dealt with before it starts by making sure horses have good experiences away from the herd. They need to bond with people so they are just as happy with people as with their horse friends. If you just do the same boring routine this can be a part of the problem. Its good to also do things like just taking them for a walk or letting them graze out side of the pasture, do something interesting to keep their mind off the other horse.

Anxiety is sometimes an issue for the horse left behind if you only have two and one is removed.
You can try to calm the left behind horse by giving it some food to keep its mind busy, and by practicing taking out the other horse and returning it after different amounts of time have gone by, this lets the left behind horse realize that its friend will come back.
There are also herbal supplements you can give to horses with anxiety issues, look for them at your local livestock feed store or tack shop.