Electricity is the flow of electrons. If the circuit is broken (by the switch on the appliance being 'off') there can be no flow, so no electricity is used.
The previous paragraph assumes, of course, that the appliance is genuinely 'off'. Most PCs, for example, which have been 'shut down' aren't genuinely switched off. (To do this you have to use the switch which is usually located on the back of the cabinet). A 'shut down' PC is often actually in a form of 'standby' mode and is therefore using a small amount of electricity. (Although the amount is small, it's been estimated that the UK could shut down at least one power station if everyone always switched off their PCs - using the rear switch - rather than just shutting them down).
Chris