Appeals are a whole 'nother world. Most attorneys don't do them, nor do most attorneys know how to do one—much less know what one really is. We, on the other hand, specialize in appeals. Because we don't have a volume practice, there's time to work on them. Plus, they're just fun. Crafting winning legal arguments is a thrill. It's combining the facts of the case with the supporting law and presenting it through good legal writing—something most lawyers just can't and don't do.
We should discuss what appeals are and are not. They are not a new or re-trial of the case. They are not a re-weighing of the evidence. The jury (or the judge in some cases) decides what the evidence is and means and says. The appellate courts exist to decide and rule on points and errors of law.
For example, an appellate court does not decide that the other side's star witness really was a liar—that's for the jury. The appellate court does decide if the witness's testimony should have been heard by the jury or if it was sufficient to support the conviction. The same is true with documents and other materials admitted into evidence. The appellate court will review if those were properly before the jury and whether they are sufficient.
If the appellate court finds error, then it will decide what the appropriate remedy is. It's not possible to discuss here what will happen because the facts of all cases and the error that occurs in them are all different. Suffice it to say that the outcome from error on appeal run the gamut from a reformation of the judgment to vacating the conviction and/or sentence and remanding the case for further proceedings to reversing a conviction and rendering the defendant not guilty in a criminal case.
Federal Criminal Appeals: 10 Things You Should Know is an excellent article on federal criminal appeals by Adrienne Urrutia Wisenberg. Almost everything in this article also applies to Texas criminal appeals.