What does the 'hp' rating actually signify, with respect to an aircraft engine?
What is horsepower? It is a measure of the work a horse can do in an hour, invented in the time of the steam engines in order to make sense of what they could do, in a time when all work was done by humans and horses. As usual before the SI, there were many definitions of horsepower. Here is the one from wiki, on what a metric horse could do:



Notice that there is a time element in the picture: if the horse takes all day to raise the weight one metre in height, he is not very powerful.

Nowadays, the unit of power is the Watt, after James Watt who coined the horsepower unit to sell his steam engines. It is defined as the force in Newton, required to move 1 kilogram over one meter in one second: 

$$1 W = 1 \frac{N \cdot m}{s} $$

In these units, you can see a force times a velocity, and that is one way of looking at what power is: you need more power to drive faster up a hill. But these are linear entities, and most power is made by rotary engines. Rotary units are torque (N*m) and how fast the engine is spinning (rad/s).

With engines, it makes a difference where power is measured. There are always losses caused by friction and by energy conversion, and the closer you measure at the output shaft of the engine, the higher the reading. This is what vehicle manufacturers rather do: car engine power is measured at the crankshaft, not at the wheels that actually propel you over the roads, because between engine and wheels there are transmission losses.

The question why power is used for engine rating, and not force or torque, is an interesting one. It is force that accelerates us: $ F = m \cdot a$. We can gear the motor force or torque output as required - if the gearing had no losses, power before and after gearing would be the same. So horsepower is useful to compare capabilities of engines, regardless of gearings. However they wil always be maximum capabilities.



With an aircraft, the engine power needs to propel the aircraft via the propeller. The propeller delivers thrust T, and can drive the plane at speed V, maximum speed being where drag equals thrust. Engine power is thrust times airspeed, plus the power required to overcome losses: mechanical losses due to friction, and aerodynamic losses due to propeller efficiency. 

Your question on having an aircraft a quarter of the weight and similar size. The thrust is required to overcome aerodynamic drag, both friction drag and induced drag from lift. Friction drag will remain equal, lift induced drag will be 1/4th of the full weight version. You will need more than 45 hp to get the quarter weight aircraft to travel at the same cruise speed.



The picture is from this most informative source: parasitic power stays the same, induced power reduces by a factor 4. Notice that at low speeds induced power is dominant, at high speeds parasitic power is dominant. So with a 45 hp engine you will get off the ground if the weight is reduced by factor 4, but the maximum speed will be way less.