Power is the rate at which energy can be utilized and work can be done. Mathematically, P = dW/dt. Power is not the same as energy; a truckload of wood and a tank of rocket fuel may have the same chemical energy, but the second can release that energy (when burned) at a much greater rate than the first, and hence provide more power.
The greatest power, however, is produced from nuclear reactions of the fission and fusion varieties, which release the binding energy of the nuclei themselves, rather than just the chemical bonds between molecules. In practice, of course, you'd want to limit the rate of the reaction by containing it in a reactor.
The SI unit of power is the Watt, defined as one Joule per second. This has almost totally replaced the old units of foot-pounds, but the power of American automobile engines is usually expressed in terms of horsepower, which can converted to SI units by 1hp = 746W. The idea of having "two hundred horses under the hood" seems "powerful" in the conventional sense and has a much greater appeal to American machismo than the geeky "kilowatts" used in Europe.
Electrical power can be expressed as current times voltage, P = IV, but voltage and current are also related by Ohm's Law,V = IR. Substitute the second equation in the first, and you get P = I2R. The power, or energy dissipated by whatever's carrying a current, is a function of the medium's resistance. Run electricity through something that's not a superconductor, and electrical energy will be converted to thermal energy and released. It'll produce heat which, depending on the situation, can be used to make toast, destroy a CPU, or turn the head of a condemned prisoner into something out of Ghost Rider. In this case at least, the political definition of a term (see item 5 in Cletus The Foetus' writeup above) very nearly matches the scientific definition.