An appellate court is a court that hears cases in which a lower court -- either a trial court or a lower-level appellate court — has already made some decision, which at least one party to the action wants to challenge based upon some legal grounds that are allowed to be appealed either by right or by leave of the appellate court. A party who files an appeal is called an appellant, and a party on the other side is an appellee or, in some jurisdictions, a respondent. Cross-appeals can also occur, when more than one party to a case is unhappy with the decision in some way, often when the winning party claims that more damages were deserved than were awarded.
An appeal as of right is one that is guaranteed by statute or some underlying constitutional or legal principle. The appellate court cannot refuse to listen to the appeal. An appeal by leave requires the appellant to move for leave to appeal; in such a situation the appellate court has the discretion to grant or refuse the appellant's demand to appeal the lower court decision.
Generally speaking the appellate court examines the record of evidence presented in the trial court and the law that the lower court applied and decides whether that decision was legally sound or not. The appellate court will typically be deferential to the lower court's findings of fact (such as whether a defendant committed a particular act), unless clearly erroneous, and so will focus on the court's application of the law to those facts (such as whether the act found by the court to have occurred fits a legal definition at issue). If the appellate court finds no defect, it "affirms" the judgment. If the appellate court does find a legal defect in the decision "below" (i.e., in the lower court), it may "modify" the ruling to correct the defect, or it may nullify ("reverse" or "vacate") the whole decision or any part of it. It may in addition send the case back ("remand" or "remit") to the lower court for further proceedings to remedy the defect. In some cases an appellate court may review a lower court decision de novo (or completely), challenging even the lower court's findings of fact. This would be the proper standard of review, for example, if the lower court resolved the case by a pre-trial summary judgment motion to dismiss which is usually based only upon written submissions to the trial court and not on any trial testimony.
Sometimes the appellate court finds a defect in the procedure the parties used in filing the appeal and dismisses the appeal without considering its merits, which has the same effect as affirming the judgment below. (This would happen, for example, if the appellant waited too long, under the appellate court's rules, to file the appeal.) In England and many other jurisdictions, however, the phrase appeal dismissed is equivalent to the U.S. term affirmed; and the phrase appeal allowed is equivalent to the U.S. term reversed.
Generally there is no trial in an appellate court, only consideration of the record of the evidence presented to the trial court and all the pre-trial and trial court proceedings are reviewed — in very rare instances new evidence may be considered on appeal if that material evidence was unavailable to a party for some very significant reason such as prosecutorial misconduct.
After copies of the record have been made and certified by the court below the appellant has the opportunity to present arguments for the granting of the appeal and the appellee (or respondent) can present arguments against it. Arguments of the parties to the appeal are presented through their appellate lawyers, if represented, or pro se if the party has not engaged legal representation. Those arguments are presented in written briefs and sometimes in oral argument to the court at a hearing. At such hearings each party is allowed a brief presentation at which the appellate judges ask questions based on their review of the record below and the submitted briefs.
It is important to note that in an adversarial system appellate courts do not have the power to review lower court decisions unless a party appeals it. Therefore if a lower court has ruled in an inproper manner or against legal precedent that judgment will stand even if it might have been overturned on appeal.