The switch is called circuit breaker and the coil is called under voltage coil.
When the supply voltage fails or too low,the coil triggers the breaker to trip. As a result, loads do not get power again when the supply voltage is restored, unless someone turn on the breaker. This mechanism is helpful for the safety of humans and machines.
Here is the pictures of the Switch, it has a level like a normal switch but it will only work when the coil internally inside the switch has voltage, no voltage on the coil the switch will not work
What kind of switch is this and what are they called?
According to picture,it is not relay or circuit breaker.
Generally you can say it "holding coil switch" or specifically you can say it "magnetically held toggle switch".
The holding coil can be used to release the switch from remote location, or it can be used to interlock some impotant conditions which must be fullfilled to start a machine safely. Besides it also can be used to provide safety against DC control power lost to restored event.