Just to chime in:
AndyM is correct in contradicting your dad. Fuel is burned to create motive force. In Newtonian physics, force = mass * acceleration. Given a constant mass for your car, the amount of fuel burned (ignoring the efficiency curve of your particular drivetrain) is proportional to the total amount of acceleration you're demanding of it. When traveling down a hill with a tailwind, you're not asking the engine to provide acceleration, while uphill (and with a trailer) that demand is greatly increased.
And mwb1100 is also correct. The "octane" reflects the amount of octane molecules; you may notice that the word sounds somewhat like "methane", "butane", "propane", etc. These are all different petroleum compounds. The octane one happens to require a greater amount of energy to stimulate its oxidation reaction. When there is more ambient energy, in particular, when the pressure is greater (remember, PV=nRT, so the compression cycle of the cylinder, where volume decreases, is pretty much equivalent to cranking up the temperature), then other petroleum molecules are prone to ignite before the cylinder head has moved to the optimal position. This early ignition (a) lowers fuel efficiency, since the explosion is fighting against the cylinder head getting down to that bottom position; and (b) increases pressure even further, creating more heat that can actually damage the engine.
In older cars, these premature ignitions can be perceived as a "knock" or "ping" sound. In newer cars, the engine has a knock sensor that will notice this before you can. In such a case, I believe the engine will compensate by changing the timing of the ignition spark, or dumping extra fuel into the cylinder in an effort to lower the temperature and stop the premature ignition -- thus lowering your fuel efficiency even more.
So, by using a fuel mixture with a higher proportion of octane molecules, you avoid this problem.
All that said, it's only an issue if the degree of compression in your engine is high enough that "normal" gas is in danger of that premature ignition. This is the case in a small minority of normally-aspirated engines (the Toyota FJ comes to mind). But if you've got forced induction (i.e., a turbocharger or supercharger) increasing the pressure, then you probably need high-octane gas.
If you don't have a car whose engine is susceptible to the problem, then paying for that extra resistance to ignition is wasted.