A hybrid is coin that has a mis-matched obverse and reverse. For example, a Roman coin of one emperor and the reverse of another emperor or family member is a hybrid. A modern day comparison would be a coin with the obverse (heads) of a US Penny and the reverse (tails) of a US Dime. Most ancient Roman hybrids are unofficial, ancient counterfeits or imitatives. Some scarce or rare hybrids are official mint strikes.
The term mule is often used synonymously with hybrid but mule should only be used to describe hybrids that are official mint issues. 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."