While the illustration above depicts a Carter 1 barrel, the theory is the same with most carburetors.
The spring mounted above the diaphragm (spring), tends to push down on the accelerator pump, called a delayer spring. At this point the throttle lever is holding the accelerator pump up. In the up position fuel is drawn from the float bowl to fill the pump reservoir. The fuel lifts up the inlet check ball which is used to block the inlet passage when accelerating. This keeps the fuel from being pushed back into the float bowl.
As the throttle is accelerating the delayer spring pushes the diaphragm down and forces the fuel out the discharge (pump jet). Here is another check ball (may be a check ball with a check weight, or a pointed check weight only). The fuel pressure lifts the discharge needle (or check ball) to allow the fuel out of the pump jet and into the carburetor throat. When not accelerating the discharge needle, or check ball blocks the passage so that the vacuum doesn't pull fuel out of the pump jet.
As the throttle is closed the accelerator pump diaphragm moves up and fuel is drawn in to fill the pump reservoir, ready for the next acceleration.
If you are having an acceleration problem, look down the carburetor bore (engine off) and pump the throttle. You should see a squirt of fuel come out of the main discharge hole.