When it comes down to pure horsepower which system is the best for generating the most power? Fuel Injector or a Carburetor?

injector!

It all depends on design.

Generally, fuel injection delivers more power, torque and economy.
It also give a more efficient "burn", as the fuel is fed into the engine under pressure.

Carburettors are different - the fuel is drawn into the engine, and the burn's less complete.

In tuning terms, whilst carbs are not as efficient as fuel injected engines, they can be made to produce much more power, dependening on the tune applied.

A big Holley carb for instance, fitted to a big V6 / V8 motor, will produce more power than the same engine fitted with fuel injection, though emissions will be higher. Carb-fed engines gave way to fuel injection for this reason.

Side draft carbs are still favoured in the tuning world today, though it's getting increasingly difficult to get them through emission tests in lots of countries, if you wish to use such a car in the state of California, at best it'll be a track car, as it'll never get through the smog test.