If the appeal court agrees to the appeal, then a number of things may be affected. Depending on the situation, a successful appeal may bring the judge to order a new trial, at which time the previous conviction is set aside and sometimes even withdrawn.
In some rare and special circumstances, when the appeal court finds that the evidence is not enough to prove the previous decision of conviction, the court may just acquit the accused person and present a finding of not guilty.
If the appeal court decides that the previous sentence was made in error, then it may choose to vary the sentence by lowering it or ordering that the sentence be served in the community.