The Jupiter Icy Moons Explorer (JUICE) is a European Space Agency mission to explore Jupiter and its three icy Galilean moons: Europa, Ganymede, and Callisto. Numerous JUICE investigations concern the magnetised space environments containing low-density populations of charged particles that surround each of these bodies. In the case of both Jupiter and Ganymede, the magnetic field generated internally produces a surrounding volume of space known as a magnetosphere. All these regions are natural laboratories where we can test and further our understanding of how such systems work, and improved knowledge of the environments around the moons of interest is important for probing sub-surface oceans that may be habitable. Here we review the magnetosphere and plasma science that will be enabled by JUICE from arrival at Jupiter in July 2031. We focus on the specific topics where the mission will push forward the boundaries of our understanding through a combination of the spacecraft trajectory through the system and the measurements that will be made by its suite of scientific instruments. Advances during the initial orbits around Jupiter will include construction of a comprehensive picture of the poorly understood region of Jupiter's magnetosphere where rigid plasma rotation with the planet breaks down, and new perspectives on how Jupiter's magnetosphere interacts with both Europa and Callisto. The later orbits around Ganymede will dramatically improve knowledge of this moon's smaller magnetosphere embedded within the larger magnetosphere of Jupiter. We conclude by outlining the high-level operational strategy that will support this broad science return.