If you have something like a light bulb then when the voltage drops so does the current and so the bulb becomes dimmer and if the voltage becomes too low the bulb will not any visible light.
A fan motor which have mentioned has a specific job to do and is designed to work at a certain voltage and current and hence with a certain power input. If the voltage drops so does the current and as there is less electrical power entering the motor the motor can do less work and so rotates at a slower rate.
However there are other electrical motors which are designed to do a variable amount of work eg household electric drills, electric trains etc. In those cases the amount of electrical power entering the motors is sometimes controlled by the motor itself. So when an electric drill is used as more demand is made on the drill the coil of the motor rotates slower and more current (and hence more electrical power) is delivered to the drill. It is not acting like a resistor. If the power supply voltage drops then that would mean that the drill will not be able to get as much electrical power and will therefore not work as well. If you have used a rechargeable drill you will have noted this effect.
Some devices eg televisions, computers, chargers etc are fairly immune to a drop in supply voltage because they have electronic circuitry which keeps the output voltage constant. If you look at a typical charger it will allow and input voltage of anything between 110 V to 240 V to give a roughly constant output voltage which is usually much less than the input voltage.
When you switch a television from standby to on the current into the television increases because a lot more of the electronic circuitry including the screen need electrical power to function.
If things are working well at home then the voltage of the supply will stay constant but as more devices are used then the supply will have to deliver more electrical current - more current is drawn.