What would make a car run well while idling but rough under heavy acceleration?

A faulty oxygen sensor can cause a car to accelerate roughly while under a heavy load. The oxygen sensors main purpose is to manage the air/fuel ratios inside of an engine. When the unit fails, the engine can run too lean, which causes the vehicle to have issues.

The oxygen sensor is located on top of the exhaust pipe. When the exhaust gasses escape from the engine, the sensor reads its air/fuel ratio. The sensor then sends the information to the car's computer so it can adjust the fuel levels in the engine. Symptoms of a bad oxygen sensor also include lower than usual gas mileage, engine pinging or detonation, erratic engine idle, engine misfire and a clogged catalytic converter.