Your authorization card indicates which types of combat you're authorized to compete in within the SCA. The list keeper must check this before you can enter a tournament, and you must have one of these to participate in any type of combat. The exception is if you are training for authorization, in which case your kingdom may require you to have a 'training card' and be under the guidance of an authorized combatant or a marshal.
In some kingdoms, authorization cards are also used for marshals and other officers to indicate that they have met the minimum standard/qualifications for the office and are sanctioned to undertake the activities of that field, e.g. first aid for a chirurgeon. In other Kingdoms, this is not necessarily the case.