What happens if a 240V appliance is connected in a 120V AC power supply?

Probably nothing serious would happen as has been mentioned already but there is a possibility of a potential serious situation. Consider an appliance only intended for operation on 240V AC but is able to work from (say) 200V to 250V. To do so might mean it uses a switch-mode power supply to regulate the internal DC voltages. Let's say it required 100 watts internally, maybe some form of audio amplifier.

At 250 volts AC it would draw 0.4 amps plus 10 % more for inefficiencies - that's a current of 440 mA. At 200 volts AC it would draw 550mA. At 100 V ac it would try and draw a current of nearly an amp if it were able.

The point is that it will try and draw more current at a lower AC voltage and this could blow an internal fuse or damage the switching transistor - the average current may only be 1 amp but the switching current might be 10 amps. Also, at a lower voltage (with the increase in current) the reservoir capacitor after the bridge rectifier will be struggling to maintain low ripple and between cycles the dc voltage before the switching element may sag to only 50 volts - this means a higher instantaneous current draw on a cyclic basis and possibly more damage to the regulating switching transistor.

There could be damage. With half the RMS AC voltage, that is, half the force pushing charge through the device, we might expect half the current flow. If the device acts like a simple resistor, that's exactly true. That means 1/4 of the normal amount of power is used by the device.

If the device has capacitive or inductive reactance, and has nonlinear effects, then no. Still, with no specific device as subject discussion, we may as well assume one quarter the power usage.

If that power is primarily working a motor, then the motor will be spinning slower. (duh.) Some motors depend on spinning at a high speed to keep themselves cool. If it's not spinning fast enough, maybe it won't keep itself as cool. But at (probably) 1/4 the power, it's not getting as hot, either. Will friction or load keep the motor from spinning at all?

Whether the cooling effect is diminished in the same proportion as the motor heating, depends on the actual type of appliance, the load the motor is pushing, the presence of voltage regulating circuits, and for all I know, the appliance's astrological birth chart.

That's just considering basic motor physics. The range of parts and physical phenomena in a generic unspecified household appliance is vast, and so it is not possible to rule out some other way that half-voltage input could cause damage.

Short answer: without further info, it's guesswork, but the range of guesses must include the possibility of damage.

There is only one way to find out, assuming you can make the plug fit the socket...

In a linear situation (an electric blanket, for example), the power will just be reduced to 25%.

Switching power supplies such as PC power supplies (the kind with a slide switch to select the input voltage) will attempt to produce the required output power with the available voltage, and unless some kind of under voltage lockout or thermal protection kicks in could be damaged - the power devices will get much hotter than normal.

The most susceptible to damage are appliances like small refrigerators that require enough motor torque to get past the compressor torque humps. With low voltage in (such as a brown-out), the compressor can stall (reducing the motor back-EMF to zero) and thus draw a much higher current than usual, all of it converted to heat. As a bonus, any cooling fans will not operate at full efficiency, if at all.