When it's working properly, the internal combustion engine is an engineering marvel. So many factors come together perfectly to make a car run and when it runs well, it is seamless. However, this same combination of many perfectly functioning parts is a double-edged sword. Any one of these factors not working properly will throw the whole thing off and can hamper performance in trivial or even serious ways, such as causing hesitation on acceleration.
One cause for hesitation on acceleration may potentially be the misfiring of your spark plugs. If one of your cylinders isn't firing at the right time (or at all) you will experience a jumpy, bumpy acceleration and engine performance. The causes for this can be many. The plug itself may be bad, there may be a touchy connection between the ignition coil and the plug, oil may have leaked into the cylinder and coated the plug.
Older catalytic converters that have seen a lot of use can actually experience meltdown. The catalyst inside will begin collapsing and fusing together. This causes exhaust gases to back up and reduces the engine's ability to produce consistent combustion because exhaust gases don't burn well. This, much like a faulty spark plug, will cause all sorts of hesitation and uneven acceleration.
One of the most common reasons for hesitation during acceleration is the failing of a fuel filter. Filters get clogged, just like an oil filter, and this clogging prevents the free flowing of gas to your engine. No gas means no explosion and no explosion means no acceleration. But clogged fuel filters will only block some of the gas some of the time or may provide enough gas just for coasting, but not enough for hard acceleration. Either way, you'll experience jerky hesitation while driving.
More serious than a clogged fuel filter, a dying fuel pump may also the cause of engine hesitation. Often the result of failing or burnt out electronics, failing fuel pumps will sputter and flail about trying to get enough fuel to the engine, but often failing. This lack of fuel of course results in spotty combustion and in turn poor or even jerky hesitation. Replace this soon as the more worn down it gets, the more stress it places on your car
Various Other Problems
As with anything as complicated as a car, there are hundred of variables that impact its performance. If you have ruled out faulty plugs, a clogged cat, clogged filter and dying fuel pump there are still other possibilities to consider. Keep in mind that a car needs air, fuel and spark to run and it is hesitating because it's not getting one of these three items consistently. Just like doctors rules out body systems when you're sick, start ruling out these systems one by one.