A 4-bit asynchronous up counter can be constructed using flip-flops, where each flip-flop represents a bit of the counter. The counter increments its value with each clock pulse, with the output of one flip-flop serving as the clock input for the next. The design can be visualized through illustrations, such as the one referenced from All About Circuits. The asynchronous nature means that not all flip-flops are triggered simultaneously, which can lead to propagation delays. Understanding the basic operation and timing of flip-flops is essential for implementing this counter design effectively.