UPDATE: THIS ANSWER IS INCORRECT. At the time of writing this answer, it was correct but the system has since evolved and this answer is now out of date. The answer posted by achapman is the more up-to-date and correct answer.
In the code they are currently called Dispute bonds, but Appeal bond is the same thing.
A dispute bond is when someone disagrees with the outcome of one of the phases of reporting (automated/early, limited reporters, all reporters) and puts some REP into escrow as a means of backing up their claim.
Anyone can post a dispute bond for a market during the dispute phase of reporting (currently 3-day window after each reporting phase). Only one dispute bond can be placed per market, and there can only be one fork per branch (universe), which means only one all-reporters dispute bond.
At the moment, the costs are hard coded to 1100 REP for the automated/early reporter dispute bond, 11,000 REP for the limited reporter dispute bond and 110,000 REP for the all-reporter dispute bond (the one that causes a fork). These numbers aren't finalized yet, but they are hopefully at least the right order of magnitude.
The goal with the bond sizes is to appropriately discourage frivolous disputes. In particular, we really want to discourage fork disputes because these cost the system a tremendous amount of work and pain. We want to ensure that if anyone does impose that on the system, they are really sure about it and if they are wrong the people most impacted are sufficiently rewarded. For the automated/early reporting, the negative impact is pretty low so the bond amount is relatively low, though still high enough to discourage most people from frivolously disputing.
Whether you get your bond back (plus bonus for taking the risk) or your bond is forfeit to reporters depends on how the market finally resolves. If the early reporter reports an outcome of Apple, this is disputed, then limited reporters report outcome Banana, this is disputed, then all reporters report Apple (which is not disputed) then the person who disputed Apple will lose their bond (because their dispute was frivolous) and the person who disputed Banana would get their bond back (plus some bonus) because their dispute was correct.