Volts is the force pushing the electricity through the coil. Amps is the rate that the electricity flows at. There's some differences, but you can think of it like water pressure and flow rate in pipes. Watts is the power that is delivered to the coil by the electricity. If the pressure is higher then the coil gets more power, and if the flow rate is higher, the coil also gets more power. Think of the difference between a fire hose and a garden hose. The former has more pressure and more water per second and it can knock you off your feet.
Resistance is like the narrowness of the pipe. The narrower the pipe, the higher the resistance, the more pressure you need to have to deliver the same number of gallons per second.
All these things are related to one another by two simple equations:
V = I x R (V is volts, I is amps and R is resistance in ohms)
P = V x I (P is watts)
You can use algebra to rearrange those. The most useful way for vaping is to get rid of current, because we don't usually know what it is.
P = V2 / R
Power heats the coil. The more watts that is delivered to the coil, the hotter it gets. But it is being cooled all the time by the vaporisation of the liquid and by the air flowing past it. The hotter it gets, the more those things cool it down. So it tends to heat up to a certain temperature and stay there, (although if you stop puffing for a second it quickly gets hotter.) The more power you supply, the quicker the coil heats up, and the hotter it gets.
Also, short thin wires heat up faster than long thick wires, even if they have the same resistance. They also have different surface areas so they cool down at different rates too.
All this is a dark art. There are only two groups of people in the world who have any idea how the temperature of a coil works: the guys at Evolv and the people working with Dr Farsalinos. But it is perfectly possible to work with it without understanding it, you just increase the watts until you get a vape that you like.
Different eliquids taste best at different temperatures, but they can all be burned if the temperature is too high. Similarly they all fail to produce vapour if the temperature is too low.
EGOs are designed to produce a good vape at around 5-6 watts. It's a matter of how quickly the wick delivers the liquid to the coil and how much air is flowing over it. You can't simply increase the power and get more vapour, you'll just burn it, and in extreme cases heat the coil so much it melts. Dual coil EGOs split the watts between two coils, so they will be best with about twice the watts. I'm currently puffing on a single-coil Russian 91% at 10-12W. Some vapers manage to build coils that will take 50 watts or more. But all of those coils are more or less at the same temperature, around 400-450F.
Varying the power allows you to do two things. Changing it slightly allows you to tune it for different liquids. Changing it a lot allows you to build coils and wicks that deliver more or less vapour.