The MINI power steering is unusual because it is "electro-hydraulic" with a fluid reservoir (central at back of engine compartment near bulkhead) but the pump providing pressurised hydraulic fluid to the steering rack, is driven by an electric motor instead of being belt driven off the engine like most other cars. The electric pump motor is what gives the distinctive whining noise on turning but I have grown to like it !
Many other small cars including the latest R56 MINI now use fully "electric servo" power assisted steering with a motor geared direct to the column or steering rack with no hydraulics or fluid reservoir and less noise.
Both types of electric power steering are used on small cars because there is no power drain* from the pump being driven by the engine and this also means marginally better fuel economy.
The system is powered by electrical current being drawn from the vehicle's electrics. They have a built-in safety device so that if the electrical power supply were to fail, then it will default to a conventional un-assisted steering system. Additionally, if the engine was to stop whilst the car is moving then the fully electrically assisted steering system will still operate, unlike a conventional hydraulic assisted system. However as some MINI owners have found out if the electric motor fails on the electro-hydraulic system the steering does suddenly get very heavy !
* Note: A few years back the smaller engined Ford Fiestas were offered with a choice of Power steering or Air Con but not both as the power losses to drive both pumps off the engine was just too great.