The small rotor on the back of a helicopter

What the tail rotor counteracts, and what happens when it stops.

The tail rotor exists purely to stop the helicopter's own body from spinning in the opposite direction to its main rotor, because spinning the main rotor one way inevitably tries to spin everything underneath it the other way, and without something actively cancelling that reaction the fuselage, not the blades overhead, would be the part doing the turning.

The physics of reaction torque

An engine driving the main rotor applies a continuous turning force, a torque, to the blades to keep them spinning against the resistance of the air. Newton's third law says that for every force one object applies to another, the second applies an equal and opposite force back, and torque is no exception: as the engine and gearbox twist the rotor one way, the rotor and the air resisting it twist the engine, gearbox and everything bolted to them the other way. The same result also falls out of angular momentum being conserved: a helicopter sitting still in the air starts with none of it, so if the engine gives the rotor blades angular momentum in one direction, the only way the total can still add up to nothing is for the fuselage to pick up an equal amount in the other direction, which is exactly the spin the tail rotor exists to cancel. In a helicopter, everything bolted to the gearbox is the fuselage, so without some separate way of cancelling that reaction, the cabin, tail and undercarriage would simply rotate beneath a rotor that itself stayed comparatively still relative to the air, which is exactly what early experimental designs without a working solution actually did. The tail rotor is a second, smaller propeller mounted at the end of a long boom, spinning in a plane roughly vertical rather than horizontal, and it is set to produce a sideways thrust specifically aimed at pushing the tail in the direction needed to cancel the fuselage's tendency to spin.

Where you have already felt this

Anyone who has felt a hand drill or a screwdriver kick back against their wrist the instant a bit suddenly bites or binds in a hole has felt exactly this reaction. The motor is trying to spin the bit, and while the bit is turning freely that effort goes entirely into cutting, but the moment it catches on something and cannot turn any further, the same effort has nowhere else to go except into twisting the tool, and with it the hand holding it, the opposite way round. A helicopter's main rotor is not catching on anything, it is spinning continuously and freely through the air, but the reaction is the same reaction all the time rather than only at the moment of catching, which is why it needs a continuous countermeasure rather than just a firm grip applied occasionally.

What follows from this

Because the tail rotor's whole job is to balance a torque, not to lift anything, its own thrust requirement scales with how hard the main rotor is working rather than with the helicopter's total weight, so pulling more power for a steep climb or a heavy load also demands more tail rotor thrust to keep the nose pointed where it should. This is also why the pedals in a helicopter's cockpit, which control the tail rotor's thrust, are being used constantly during a flight rather than only when turning, since every change in main rotor power changes the reaction torque that needs cancelling and the pilot has to adjust for it almost continuously. If the tail rotor fails outright, whether from mechanical damage or a broken drive shaft, the fuselage is left with nothing opposing the main rotor's reaction, and it begins to spin, uncontrollably and increasingly fast, in the direction opposite the rotor overhead, an emergency that pilots train for specifically because it follows directly from how the whole arrangement works in the first place, and not from any single part actually breaking.