A mule is an official mint issue Roman coin that is a hybrid, meaning it has the obverse and reverse type that do not belong together. Frequently hybrids will have am obverse of one emperor and the reverse of another. The numismatic term mule is derived from the animal mule, the hybrid offspring of a horse and a donkey, due to such a coin having two sides intended for different coins, much as a mule has parents of two different species. Coins of this type are generally scarce or rare. A modern day comparison would be the obverse (heads) of a US Penny and the reverse (tails) of a US Dime. Unofficial, ancient counterfeit or imitative, hybrids are very common and should not be described as mules. Curtis Clay explained on the Classical Numismatic Discussion Board:
"The term "hybrid" has become tainted in my mind, because so many of the coins so described in the earlier RIC volumes and in Roman Silver Coins are nothing but ancient counterfeits. Therefore I prefer to say "mule" for coins struck at the mint from mismatched official dies. The distinction official/unofficial is crucial. Official mules are for the most part very rare, and interesting as error coins and for showing a chronological connection between dies that we otherwise wouldn't have known were in use at the same time. Unofficial hybrids are very common and teach us nothing about the chronology of the official coinage."
Official mules are related to either contemporaneous or consecutive issues. For a contemporaneous issue mule error, a reverse die is used for the wrong emperor in an issue that struck coins for both rulers. For example. an obverse of Septimius Severus, combined with a reverse type of Caracalla from the same issue. For a consecutive issue mule error, hybrids join one die showing a new, recently introduced type or legend, with a die of the preceding issue that had erroneously or carelessly remained in use, showing a now superseded type or legend. For example a denarius of Caracalla with TR P II on the reverse, coupled with an obverse die of the preceding issue, still showing just TR P. Consecutive issue mule errors also include coins from the first issue of an emperor, struck with a reverse die from the last issue of the previous emperor.
Hybrids that are not either contemporaneous or consecutive are always (or almost always?) unofficial.