One of the most important ideas in all of science is the law of conservation of energy: energy can neither be created nor destroyed; it can only be transformed from one form to another. The total energy of a system (and of the whole universe) stays constant. When energy seems to disappear, it has actually changed into another form — often heat or sound that is hard to notice.
The clearest school example is a freely falling body. Imagine a ball dropped from a height $h$. At the top it is at rest, so its kinetic energy is zero and all its energy is potential, $E_p=mgh$. As it falls, its height decreases (so $E_p$ falls) while its speed increases (so $E_k$ rises). At every point the sum stays the same:
$E_p+E_k=\text{constant}$, i.e. $mgh+\frac{1}{2}mv^2=mgh_{\text{top}}$
Just before hitting the ground all the potential energy has become kinetic energy. A simple pendulum shows the same idea repeating: at the highest point of its swing it has maximum potential energy and zero kinetic energy, while at the lowest point it has maximum kinetic energy and minimum potential energy. Energy keeps shuttling between $E_p$ and $E_k$ (in real life friction and air resistance slowly convert some into heat, so the swing dies down).
Power tells us how fast work is done or how fast energy is transferred. Two cranes may lift the same load to the same height — doing the same work — but the one that does it in less time is more powerful. Power is defined as the rate of doing work:
$P=\frac{W}{t}$
where $W$ is the work done (or energy transferred) and $t$ is the time taken. The SI unit of power is the watt (W), named after James Watt. One watt is one joule of work done in one second, so $1\,\text{W}=1\,\text{J/s}$. Larger units are the kilowatt ($1\,\text{kW}=1000\,\text{W}$) and the megawatt ($1\,\text{MW}=10^6\,\text{W}$). The older unit horsepower is roughly $746\,\text{W}$.
For everyday electricity bills, the joule is too small, so we use a much larger commercial unit of energy: the kilowatt-hour, written $\text{kWh}$ and often called one ‘unit’ of electricity. One kilowatt-hour is the energy used by a 1 kW appliance running for 1 hour. Converting to joules: $1\,\text{kWh}=1000\,\text{W}\times3600\,\text{s}=3.6\times10^6\,\text{J}$. So a 1000 W heater used for one hour consumes exactly 1 kWh of energy.