Its a long story... But the most efficient way to distribute electricity in quantity is in three "phases." (That is why you always see a multiple of 3 wires on transmission lines. )
Voltage is "Stepped up" by a transformer for distribution over long distances, then "Stepped down" at the end user.
These three phases are derived by three windings in a generator, 120 degrees apart. Usually, the coils are connected in a "Y" configuration. One end of all three coils are connected together, and grounded. The voltage between wires is a vector sum, that is why you have 208/120 V.
Apartment buildings typically have use all three phases, so their transformers supply 208 volts. Each individual apartment will have one "phase," with 208 volts.
A single family home uses only two of the three phase wires. The transformer changes the voltage to 240/120. Since all the voltage is the same phase angle, the vectors are in the same direction... 120 + 120 = 240.
Clear as mud?
In the US, most household appliances are rated 240 volts, so they may be used in single family homes.
The difference between how an appliance operates on 208 vs. 240 volts depends on whether it is a heating (resistive) type appliance like a dryer, oven, or water heater, or a motor driven appliance, like an air conditioner. Both will be less efficient when connected to 208 volts, but in different ways.
A heating appliance will simply deliver less heat. A motor will deliver the same horsepower, but will draw more current.
Many appliances like dryers and ranges also have 120 volt control systems. Since there is always 120 volts available, the system has no effect on the controls.