Poor acceleration in a car is often a symptom of engine hesitation. Unfortunately, engine hesitation has many possible causes, ranging from a simple dirty air filter to a serious transmission problem.
Air and Fuel Filters
If your car's air filter is plugged, it prevents the proper amount of air getting to the engine, which could cause engine hesitation. The air filter is simple to check, since it's usually under the bonnet. Similarly, a clogged fuel filter doesn't allow adequate fuel pressure into your car's fuel injection system. Blowing air through the fuel filter will tell you if the filter is clogged.
Spark Plug Problems
The problem may have arisen from your car's spark plugs being worn or adjusted incorrectly. Proper timing is crucial to engine function, and if this one component produces improperly timed or weak sparks, it may be enough to cause engine hesitation.
Your engine may be fine but if the power isn't effectively getting to your wheels, you may have a transmission problem. Such a problem could cause acceleration issues when the transmission slips, because it causes the car to lose the power the engine just generated, while the transmission would normally transfer the power to the rest of the car.