The overflow tube does something similar to its name— it ensures that the excess water from the tank does not overflow and dump onto the floor surrounding the toilet. The overflow tube is housed inside of the tank and usually runs vertically if installed correctly. This part is vital to the toilet functioning, and actually does its part in case the toilet malfunctions. If the ballcock in the toilet were to malfunction or break, the overflow tube prevents the tank from overflowing.
This piece works closest with the refill tube. The refill tube is another tube part located right above the overflow tube in an arch shape. In order for the overflow tube to work properly, the refill tube must be sitting above the overflow, not jammed down inside the overflow tube, instead with both of the tubes touching slightly. The overflow tube is one of the largest tube pieces within the toilet, and functions as one of the smartest toilet pieces.