Put simply,
High specific power output engines are stressed, often needing exotic materials, complex systems, narrow fuel parameters, lengthy design & test, etc and as a result are expensive. They also tend to be less reliable and more tempermental.
Basically, torque is what you're primarily interested in, it expresses the ability of an engine to turn a prop against resistance. High torque outputs are achieved by using a long stroke, which mitigates against a high reving engine due to reciprocating mass. Best examples are large diesels, huge torque but low power to cc ratio.
Torque, btw, is the only "measured" output of an engine. Power (work done) is calculated by measuring the force moved over a distance (torque) by the time involved.
There are many texts on the subject of engine output, key issues are things like mechanical and thermal efficiency, mean effective pressure etc etc.
Apologies if I've over simplified the issue, not sure what level you're coming from