Both of you are right actually;
DoCMT is correct your
alternator normally provides all the required power to operate the vehicle AFTER it has been started. The battery is mainly for starting the vehicle as this will draw the most amperage out of any operation of the vehicle. BUT, the second thing an alternator does is recharge the battery's lost charge when the vehicle is cranked over.
The key idea there is that an Alternator and the Regular Auto Battery are not designed to Deep Cycle...
Deep Cycling is drawing a battery to a low charge level, (in the case of automotive batteries is it is below 9.5volts it is often considered a "dead battery") and the charging it back to a full charge status repeatedly.
As a rough rule of thumb it takes roughly an hour worth of driving to recharge the battery the amount you drained from cranking it over. Now imagine your alternator trying to keep your car running AND recharging a dead battery, it literally burns itself out. The alternator is not designed to hand that much of a load, its meant for essentially operating the vehicle and topping its battery off not recharging it from zero.
DCDude is also right, pretty much any dead/weak battery will not usually start a car, though that is why people get jump starts.
Sorry if its written rather jittery like, trying to keep it concise without going into how the battery operates as well as the charging and starting systems in too much depth. (Though I guess if you'd like I could knock it out if people really want to know? :shrug