The other side is appealing, what does that mean?

An appeal means that one of the parties is requesting that the decision in your case be looked at again by a higher court. In other words, they are arguing that something went wrong with the first decision and that it should be changed. In most cases, the appeals process is somewhat limited. You probably aren't going to have to go to court again, and the appellate court (the one reviewing the decision) is probably going to give a lot of deference to the already-standing decision. 

Generally, when a case is appealed, there must have been a legal or procedural error for the court to change the decision. In other words, the appellate court usually isn't going to disturb decisions that have been made on the facts. Instead, they will look to see whether the law was applied correctly and whether everything was done right. If a jury was given bad instructions, if the judge let evidence in that he shouldn't have, or if something else of that nature went wrong, then the case may be “overturned,” meaning, the original decision will be changed, or it may be remanded, which means it will be sent back to the lower court to look at again with more explanation. 

This isn't to say that an appeals court will never look at a case with fresh eyes. When they do, it is called "de novo" review, which means that they look at everything with no eyes and don't pay attention to what the court below did. Usually, only decisions made by judges in certain courts go through a "de novo" appeal, like decisions in small claims court. 

In any case, usually the appeals process just consists of lawyers writing briefs or “arguments” about why the higher court should or should not change what was decided. It is also important to note that either party can appeal a case. If the winning party isn't satisfied with the amount of damages for example, he may appeal. The only exception is a prosecutor in a criminal case, who is not allowed to appeal a verdict of not guilty. 

The appeals process is legally complex. If you are involved in a case and the other side is appealing, you need to get legal help from an experienced attorney.