Is there such a thing as an appeal in Jewish law, or are rulings issued by Jewish courts absolutely final?
There are appeals, but only in some cases:
From the pasuk containing וְנָקִי וְצַדִּיק אַל-תַּהֲרֹג, we learn that you do not punish an acquitted person- even if we find out he is not righteous. We also do not punish a righteous man, even if he wasn't acquitted. Sanhedrin 33b
So in punitive cases, guilty verdicts may be appealed (by anyone, not just the defendant). Not-guilty verdicts may not be appealed.
Monetary cases can be appealed (by anyone) under certain circumstances, while in others, the judge may or may not be liable for his misjudgment. The halachos and disputes therein are complex and I haven't seen them for a few years, so I won't try to summarize. You can find them in Shulchan Aruch C.M. 25.
About monetary matters there is a quote in the gemarah: "Bei Dina Basar Bei dina lo daiki". (A court doesn't look after another court). The reason given is that if one could, there would be no end to the matter (one would go around looking for a court that agrees with him) and because we have an assumption that court rules correctly. Nowadays, many poskim say that when one goes to a court where both litigants and the judges themselves know that there is an appeals process, then there is no problem with an appeal.
However, not all agree (R' Sonnenfeld was in opposition to such a proposal and was one of the reasons that the Eida Hachareidis broke off from the Rabbanut)