Let's start off with Power.
Power in Physics is a measure of the rate at which Work (or energy consumed/transmitted) is done. It is found by the formula P = work (or energy)/time and the units are Watts. In the real world it means that a 100 Watt light bulb transforms 100 Joules of electrical energy into Heat and Light every second.
As reported in other answers, electrical Power can be found by the equation Power = Current x Voltage, but this comes from the fact that Electrical Energy = Voltage x Current x Time (E = VIt) and so substituting in P = E/t we end up with P = VIt/t, the 't' crosses out and we end up with P=IV. Then, by Ohm's Law (V=IR) this formula can be re-written as P=V*2/R or P=I*2R [note I*2 means the square of I]. So, though Power is indeed the product of Voltage and Current, it still is - like all Power - the rate at which work is done (or rate of energy consumed).
Current (symbol I) is a measure of the amount of Charge (1 Coulomb of charge is equal to 6 x 10*18 electrons) passing through a point in the wire each second. So a current of 2A (unit of Current is the ampere [A]) means that 2 coulombs of charge are passing through that point each second. So Current = Charge/time (I = Q/t).
Voltage is a measure of the work (or electrical energy given) to each coulomb of electricity passing from one point of a circuit to another. So, a 1.5 Volt battery gives 1.5 joules of electrical energy to every coulomb (6 x 10*18 electrons) of electricity. The negative side of the battery gives this energy and the electrons must lose this energy by the time they reach the battery's positive side. This usually happens by operating a device of some sort, like a bulb which transforms this energy into heat and light. If there is no such device to transform the energy into, the electrons lose all their energy as heat, like a heating coil does. That is why a short circuit is very dangerous and could start fires!