What is a bitcoin fork example?

Could you elaborate on what constitutes a bitcoin fork, providing a concrete example? In the cryptocurrency world, forks are often discussed, but for those less familiar with the intricacies of blockchain technology, a clear and concise explanation paired with a specific instance would be highly beneficial. Could you outline the basic premise of a fork, explain its potential implications, and then illustrate this with a real-world scenario that involved a bitcoin fork? Your insights would be invaluable for those seeking to deepen their understanding of this complex yet fascinating phenomenon.