Low-Earth orbit (LEO) satellite beam-hopping (BH) technology is emerging as a promising approach to meet the ever-increasing global connectivity demands, enabling agile, on-demand coverage. LEO satellite BH can address the spatio-temporal non-uniformity of ground user traffic by dynamically allocating capacity and optimizing network performance. Cooperative multi-satellite BH enables joint transmission and interference avoidance to improve received signal quality. This article provides a comprehensive paradigm of BH, detailing its key dimensions, strategies, and architectures. Through exploration of key challenges, including beam pattern design, on-demand scheduling, and interference management, this paper identifies the potential applications of BH, ranging from adaptive capacity allocation for hotspot areas, low-power Internet-of-Things (IoT), delay-sensitive services, to massive connectivity support. Furthermore, a system-level analysis is presented, including key metrics, models of inter-beam and inter-satellite interference, and cooperative joint transmission, and a case study is provided to demonstrate the performance benefits of BH with cooperative transmission. Several promising future research directions are discussed to guide the future development of LEO satellite BH networks.