Contacts are the conductive pieces of material that form part of an electrically controlled switch. Auxiliary contacts form part of a contactor, which is a type of switch that can be controlled remotely.
Contactors are often used in very high-power circuits, and auxiliary contacts are the last phase of contact, used to reduce the power of the current. They have a lower current rating than regular power contacts, and can be used to de-energise the contactor.
Auxiliary contacts are affixed to the main contacts and change state when the main contacts do. They pass the data from the circuit to a logic controller, which in turn gives instructions to the linked devices about whether to turn on or off.
Alternatively, they can be used as a standalone hardware control unit.
One common usage of auxiliary contacts is in push button controllers, where they can be built in to the start button.