Because that is such a difficult threshold, most appeals assert legal errors. That might be that the judge gave the jury wrong instructions on what the legal standards are, that it misintrpreted the law in rendering a decision on some point in the case, or that erred in admitting evidence. Evidentiary decisions are usually considered discretionary so unless the defendant can show that the trial court got the law wrong when it admitted or excluded the evidence, the reviewing court will again usually defer to the trial court's decision.
There also are other grounds for appeal like prosecutorial misconduct (e.g., withholding evidence, making improper argument to the jury), juror misconduct (jurors are only supposed to decide the case after all the evidence is heard and based only on what is introduced at trial, but sometimes violate those standards), newly discovered evidence (this usually has to be brought to the trial court first), and ineffective assistance of counsel.
Also, Prancer used the phrase "prejudicial error." That means that it is not enough to just identify some error that was made in the trial court. The error has to be shown to be "prejudicial," i.e., the defendant has to show that if the decision had gone the other way, the outcome of the case would likely have been different. Appellate courts often will acknowledge that a trial court made a mistake, but will deem it "harmless error."