Apparent power = V x I

Real power = integral(V(t)*I(t)*dt, 0..t) / t = V x I x cos(phi) (for sinusoidal signals), where phi is the phase shift between voltage and current.

Calculating reactive power is left as an exercise for the student

. For phi = 0, real power = apparent power, and reactive power is zero.

Think of reactive power as the power drawn by an ideal capacitor: there is voltage over the cap, current flows through the cap, but the ideal cap does not get hot at all. This is because voltage and power are 90 degrees out of phase. Inductive or capacitive load tends to cause this. Power factor correction (PFC), often used in power supplies which can deliver large amounts of power (eg. computer PSU), is designed to compensate for this.

Real power is the sum of the instantaneous product voltage and current, i.e. the sum of the apparent power at each moment in time.

Consumers pay for real power to their utility, commercial customers tend to be charged if they draw large amounts of reactive power, because it causes loss for the utility (that current is going to cause I

^{2}R losses, which is real power).

I'm sure you'll find a nice explanation on Wikipedia or any random electronics textbook.