Every internal combustion engine requires a certain mixture of air and fuel to operate correctly. Through years of research and development, auto manufacturers decided that this optimum ratio is 14.7 to 1 for stock applications. That means that for every 14.7 parts of air the engine takes in, there must be 1 part fuel for the engine to perform at its best. Less fuel creates a lean condition; more fuel creates a rich condition. A lean condition gives a bit more power, but also results in massive heat accumulation. A rich condition robs the engine of power, but results in cooler running temperatures. If the engine is too far off in either direction, the vehicle might not be able to pass an emissions test and might not run well.
If the oxygen sensor goes bad and tells the computer that the engine needs more fuel when it in fact does not, the vehicle will run rich. Telltale signs of a rich-running engine are a noticeable odour of gasoline coming from the tailpipe, erratic idle, fouled spark plugs and the aforementioned loss of power. Although the computer actually does all the air and fuel adjustments, it is the oxygen sensor's job to tell it which way to go. Since a rich condition robs a vehicle of power, a bad oxygen sensor can cause poor acceleration. If your vehicle is running rich, replace the oxygen sensor to see if that solves the problem.
Another Possible Issue
While the oxygen sensor can indeed cause an engine to run poorly, slow acceleration could also be a clogged catalytic converter, or CAT. The CAT is located further down the exhaust pipe from the oxygen sensor. Its job is to superheat the exhaust fumes before they reach the tailpipe, so that the result is cleaner emissions. Over time they clog up and cease to work properly, which can result in poor acceleration, an erratic idle and failed emissions tests. In a new, clean-burning engine, the CAT will perform effortlessly and do its job. However, if the oxygen sensor goes bad, which may result in more emissions, the CAT can fail prematurely, so the two problems can actually run hand in hand.