Any person who has been convicted of a crime can appeal the decision to convict if they believe a legal error has occurred. It’s a popular misconception that any case can be appealed, but there actually has to be legal basis to do so.
When a case is being appealed, the former defendant, now referred to as the appellant, will find an attorney and have them file a notice of appeal to a higher court, the appellate court, which includes a written summary of their argument toward the appeal and asks for them to review the record and determine whether legal errors occurred. This process can be long, since the party who is being appealed must send a written response to the claims, and then the appellant can respond to them in writing before the case moves forward.
From there, the appeals court will decide if the written briefs are enough, or if they need to hear an oral argument in order to decide the case. The appellate court takes in to account whether or not legal errors did occur as a part of the original case, and if there was in fact a legal error, was it one that really altered the outcome of the case.
Next, the decision goes one of 2 ways: the appellate court upholds the original ruling and denies the appeal, or the appellate court will reverse the original judgment and determine what the next steps are. This could include modifying the original judgment, asking the court to reconsider the original facts and new evidence, or even beginning a brand new trial, which means the appellant needs to again find an attorney.
Appeal lawyers specialize in knowing whether or not a case was decided incorrectly, so someone who was convicted should take care to find an appeal lawyer. Law firms for appeals, and especially lawyers for criminal appeals are a pretty specialized group, so some research can be required when someone is looking for top appeal attorneys.
Good references here.