A crime is an act which violates law. Crimes change as laws change, but certain things are always considered crimes, such as theft, rape and murder. Note that the definitions of of all these things changed depending on the culture and time period.
In period, certain crimes could be atoned for with cash payments, such as the Norse practice of weregild, the payment of a blood-price when someone was killed, thus preventing a feud. Other offences, primarily religious ones, could be atoned for by going on a pilgrimage.
Many (if not most) of the punishments employed by the law in period would be considered cruel and unusual today. Imprisonment in dark, enclosed dungeons (or worse, an oubliette), the use of stocks or the tumbril, exile, torture, maiming and execution were all commonly used punishments.
Crime and Punishment in the SCA
Participants at SCA events are expected to obey all the twenty-first-century laws of their country, even if such laws would not have been period. Over and above the general issues of respect for fellow SCAdians, if the SCA is held even partly responsible for an offense, insurance rates (and hence site fees) would skyrocket and ruin the game for many.
Punishments in the SCA for criminal behaviours are limited, but can include banishment in serious cases.