An appeal is the process of asking the courts for a formal change to a decision. That is the only thing that is the same about what is involved in an appeal, as the procedures for doing so vary from jurisdiction to jurisdiction and depending on what kind of a case is involved.
The court that hears an appeal is referred to as an appellate court. Plaintiffs get to this court largely because they are not happy with the decision rendered in their case in a lower court. While an appeal is not available in all cases, that route is sometimes open to a plaintiff on specific grounds. Those grounds could include errors in due process, fact, or law.
There are two kinds of appeals: appeal as of right and appeal by leave (permission). Appeals as of right are guaranteed by statute or constitutional or legal principle, and the court must hear them. Appeals by leave or permission mean the appellant has to get permission to appeal, and this request may be granted or refused. The U.S. Supreme Court only hears appeals if the court so agrees in its discretion, and grants a writ of certiorari.