To get a high sensitivity many relays use a polarising magnet, so that the core is almost at the point where it will pull in, and the extra current in the coil drives it to the point where it pulls in. Remove the current and there is just not enough remnant magnetism to keep it closed, and it opens. Have a slightly stronger magnet and then you have a bistable relay, the little extra current on the coil in one direction either adds or removes from the field in the magnetic circuit, and the armature then will move to either the open or closed state, and will be stable there providing there is low enough vibration and mechanical shock.
Others have 2 coils, wound on the same, or opposite sides of the armature, and use a small high remnance magnet (so that it does not lose field strength with time and the coil being powered for long periods) to provide the latching action needed.
There is a further bistable relay that has no magnet, using instead a mechanical switch arrangement that rotates a small pawl around on a shaft, and the pawl drives the contacts so that a pulse will make it change state. A variant has a separate reset, and a pawl with 3 detents, so that you have a brief contact open on energising the coil, and after 3 such the contacts stay open, needing a mechanical reset to release them. These are generally used on power lines, so that a brief overcurrent event ( branch falls on line and shorts it out while being fried off, or your furry rodent bites the cable) disconnects power to stop any arcing, but if there is a permanent fault it will stay open and not power the line till reset by a maintenance crew.
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