How do power and thrust curves compare?
Aircraft power curve
(a.k.a “polar”—a completely illogical name because somebody long ago plotted them in polar coordinates, which makes no sense, but the name stuck)
This is just the power balance of the aircraft: the power provided by the propulsion system minus the power taken away by drag.
You can divide the values by velocity (true airspeed) to get a force (thrust - drag) curve instead. But usually just the power curve is plotted and lines through origin are used to read the interesting force maxima.
Two useful observations can be made about meaning of the power and force here:

Excess power is linearly proportional to sustainable climb rate, simply by $P = m g v_v$ (where $P$ is power, $m$ is mass of the aircraft, $g$ is gravity and $v_v$ is vertical velocity).

Excess thrust is linearly proportional to sustainable angle of climb. Well, actually there is some tricky trigonometry involved, but at low angles (non-aerobatic aircraft only do very shallow climbs and descents, below about 10°) you can approximate $\sin x \approx x$ and $cos x \approx 1$ and say that $T = m g \gamma$ (where $T$ is thrust, $\gamma$ is flight path angle and $m$ and $g$ as above).


This is true independent of what kind of propulsion, if any, the aircraft has. Power curves work the same for propeller, jet and rocket aircraft and gliders alike. Just the function describing power available for given speed differs by the propulsion type.
Kinds of power
A propulsion system is a device that converts one form of energy to another with some efficiency. So in has input power ($P_{in}$)—which is the rate at which energy comes in—and output power ($P_{out}$)—which is the rate at which it provides the desired energy. They are related by efficiency ($\eta$: $P_{out} = \eta P_{in}$). The rest of the energy—because energy is always conserved—is wasted (often as heat, but in aircraft propulsion the energy given to the reaction mass, the air, is also wasted).
Powered aircraft usually use internal combustion engines consuming hydrocarbon fuel, so the input energy is fuel flow times heating value of the fuel.
The output power of the whole propulsion system is then the energy given to the aircraft (to replace the energy taken away by drag), and this is always thrust times velocity. They are related by the propulsive efficiency, which varies with velocity, and varies differently for different propulsion systems.
When propeller is used, you can split the system in two parts, with their own separate efficiencies: The engine burns fuel, drives the shaft, and wastes the rest of energy as heat of the exhaust gases. It's output power is the torque times shaft speed (the RPM). The propeller then takes this power as input and accelerates some air to produce thrust. But doing so it has to sacrifice some energy to that air (its kinetic energy increases), so its output power is lower than that of the engine.
Note to units

Power=force x velocity, and we can apply that in different ways. If we were talking about electricity we could say Power is measured in Watts, and Watts= Voltage x Current. Because we are talking about prop aircraft, we should use a power unit like HP (for reciprocating engines). HP= Torque X RPM.

Power is always the same dimension, and therefore its basic unit is always Watt. This is a derived unit composed as $\mathrm{W} = \frac{\mathrm{J}}{\mathrm{s}} = \frac{\mathrm{kg}\ \mathrm{m}^2}{\mathrm{s}^3}$. Power is voltage times current for electricity, and the units work out, because $\mathrm{V}$ is just $\frac{\mathrm{kg}\ \mathrm{m}^2}{\mathrm{A}\ \mathrm{s}^3}$ (and Ampere is a base unit).
For fuel, power is heat value, in $\frac{\mathrm{J}}{\mathrm{kg}}$, times flow rate, in $\frac{\mathrm{kg}}{\mathrm{s}}$ and we are at Watt again.
And of course torque in $\frac{\mathrm{kg}\ \mathrm{m}^2}{\mathrm{s}^2}$ times angular velocity in $\frac{1}{\mathrm{s}}$ gives you Watt again. (watch out here; it requires angular velocity, in radians per second, not revolutions per second, but radian works as dimensionless here (metre per metre), so this is prone to error by $2\pi$ that dimensional analysis won't catch).
A horsepower is not an independent unit, just a funny multiple of Watt. Multiple different multiples of Watt, actually, because there are some variants.
Engine power as function of velocity
Piston engine power grows with RPM, then plateaus (at the expense of declining efficiency) and then you hit the maximum permitted RPM.
A constant speed propeller allows running the engine at the optimal RPM, and it's efficiency only decreases slowly with airspeed. Therefore a piston engine with constant-speed propeller produces relatively constant power over the design range of the aircraft, while the thrust declines about inversely proportionally to velocity. Thus it makes most sense to give power as the main figure describing it.
With fixed pitch propeller, you are limited by not being able to reach the optimal RPM at low speeds, and by having to throttle the engine to avoid exceeding the maximum RPM at high speeds, so the power declines faster as you get away from some optimal design speed. That's why fixed propellers are only used on slower planes where it does not matter that much. The power curve is still flatter than the thrust curve (dividing by velocity), so it still makes more sense to give power as the main figure though.
With turbojet engines on the other hand the available thrust remains roughly the same. It first declines with speed somewhat similarly to a propeller, but then the ram pressure starts increasing the effective pressure ratio and the available thrust starts growing again. That's why jet engines are generally rated with thrust, not power. But it isn't really constant over speed, just like the power isn't really constant for a piston engine and propeller.
And this carries over to turbofan engines, that are actually somewhere between. Their thrust decreases with speed, though slower then for propellers, and their efficiency increases. It is an oversimplification to rate them with just static thrust, but it remains the primary value given.