What does it mean if an appellate court affirms a judgment?

"Affirmed" means the appeals court found no error or cause to reverse the decision of the lower court, and upheld the original decision. This means the decision of the lower court is final, unless the losing party petitions a higher court of appeal (if one exists). If the highest court capable of reviewing the case affirms the lower court decision, the decision is binding and the case legal complete (res judicata).
When an appellate court 'affirms' the decision, they are agreeing with the lower court.

As an example, let's say:

Matt is arrested and charged with first degree murder. During the trial, it comes out that the police did not read Matt his Miranda Rights. If Matt is convicted of murder, he has the option to appeal to a higher court.

Matt appeals over the fact that he was not read his rights after arrest. If the appellate court 'affirms' the previous court's decision, Matt's appeal was dismissed (or Matt lost).

Matt would still be forced to serve the first degree murder sentence.