Engines and people have a lot in common; both inhale oxygen, and both use that oxygen to burn hydrocarbon fuel -- the only real difference is that cars burn gasoline at very high temperature, and humans burn sugar at very low temperature. But much like that meatloaf in your fridge, gasoline can "go bad" when left to sit. Gasoline is after all, essentially just like liquified animal -- and that's not going to stay fresh forever.
A Case of Bad Gas
Gasoline breaks down in several different ways, and for different reasons. Gasoline isn't a single substance; it's composed of several different hydrocarbons. Some of them are designed to be burned in the engine, while others are designed to act as solvents to keep the first bit in liquid form. Those solvents eventually "break" the other hydrocarbon molecules, rendering them useless as fuel. Gasoline also oxidizes -- burns -- in the presence of oxygen, a process identical to but much slower than what happens in your engine. The ethanol in modern gasoline also absorbs water in the air, which isn't particularly conducive to horsepower or fuel economy.
Horsepower and Fuel Economy
Bad gas doesn't burn well; it catches fire when it isn't supposed to, doesn't burn evenly when it does catch, and burns slowly after that. So horsepower and torque are sure to suffer when an engine uses bad fuel. In the mildest scenario, the engine may just feel a bit sluggish, as a significant portion of the fuel leaves the cylinder without contributing any energy to movement. In more severe cases, the car will hesitate, buck or jerk while accelerating, just as it would if the car were running on fumes. Which, essentially, it is. That fuel that leaves your cylinders without getting burned isn't going back to the tank, so you end up spewing dollar bills from the tailpipe.
More Serious Problems
Because bad gas doesn't burn well or evenly, its octane and cetane ratings are effectively far lower than they should be. Modern engines designed to run on 87-octane fuel aren't going to be too happy with gas that acts as though its never seen the high side of 40 octane; expect knock and ping under acceleration, and a concordant loss of power. The best-case scenario here is that your computer's knock sensors notice the detonation and retard timing and decrease the air-to-fuel ratio; in which case, you'll only notice a check-engine light and a loss of power and fuel economy. The worst-case scenario is that components in your engine that can melt, break or explode do exactly that.
What You Can Do
In this case, an ounce of prevention is literally worth a gallon of cure. If you have to store a car with gasoline in the tank, a few ounces of fuel stabilizer will go a long way toward warding off damaging oxidation for a year or so. Just remember to drive the car for a few miles after dropping stabilizer into the tank, so the stabilized fuel can work its way through your fuel system and injectors. If you know you've got bad gas -- which reeks of acetone, making it hard to miss -- then the smartest thing to do is to drain your fuel system and get rid of it. You can try bringing it back with additives, but it's not worth risking your engine to save a few dollars worth of fuel.