The ARIES-CS study focusing on the conceptual design and assessment of a compact stellarator power plant identified the important advantages and key issues associated with such a design. The coil configuration and structural support approach represent key design challenges, with the final design and material choices affected by a number of material and geometry constraints. This paper describes the design configuration and analysis and material choices for the ARIES-CS magnets and its structure. To meet aggressive cost and assembly/maintenance goals, the magnets are designed as lifetime components. Due to the very complex geometry, one of the goals of the study was to provide a robust operational design. This decision has significant implications on cost and manufacturing requirements. Concepts with both conventional and advanced superconductors have been explored. The coil structure design approach adopted is to wind all six modular coils of one field period in grooves in one monolithic coil structural shell (one per field period). The coil structural shells are then bolted together to form a strong structural shell to react the net radial forces. Extensive engineering analyses of the coil system have been performed using ANSYS shell and solid modeling. These include electromagnetic (EM) analyses to calculate the magnetic fields and EM forces and structural analyses to evaluate the structural responses and optimize the coil support system, which has a considerable impact on the cost of the ARIES-CS power plant.