What causes a car to hesitate when it accelerates?

Fuel injected cars have several sensors monitoring engine rpm, engine load, speed and throttle position allowing a computer to decide the correct amount of fuel for smooth acceleration..

Carbureted cars have the throttle connected more directly and simply dump fuel whenever the pedal is pressed, simple but effective.

ADDED: The answer is a little more complicated than the 2nd scenario and would take up much more space to explain. Suffice it to say that you should try running a can or two of carburetor cleaner through it via the gas tank, AND get a spray can of carb cleaner and thoroughly spray down the throttle linkage where it is mounted on the side of the carb to make sure that everything moves smoothly. See what happens after you've done this, and if it still is exhibitting the same problem take it to a mechanic that still knows how to work on and/or adjust carburetors, because iyou could have internal problems that might require a rebuild.

Not enough fuel, maybe because the fuel pump is not putting out enough pressure, or more likely and cheaper. Replace the fuel filter that is just above the transmission on the firewall.