In a civil action, the appellee is the person that is defending themselves against an action being taken against them in the appeals court. The appellee can be either the plaintiff or the defendant from the original action. Once an appeal is brought, the person who agrees with the initial decision of the lower court is appellee.
The appellant takes a case and the appellee is assigned by default of being on the other side. It is probably that the appellee won the first court action and is the reason why the appeal is taking place. In effect, a plaintiff in the first action can turn into the defendant in a second action on the same matter.