Collective spin systems in solid-state materials are promising platforms for quantum-enhanced sensing. In systems such as nitrogen-vacancy centers in diamond, donor spins in silicon, and magnetic materials, interactions among many spins can generate collective quantum correlations that improve measurement precision beyond the standard quantum limit. This tutorial develops an intuitive, materials-oriented framework for understanding these effects through nonlinear collective-spin dynamics. We show how nonlinear interactions reshape collective spin states in phase space, redistribute quantum fluctuations, and generate metrologically useful states such as spin-squeezed and entangled states. We then connect these ideal collective dynamics to microscopic interactions and realistic solid-state systems, discussing how interaction strength, disorder, decoherence, and experimental control affect sensing performance. Different platforms and experiments are compared to highlight key material properties, advantages, and limitations. The tutorial focuses on a materials-oriented understanding of nonlinear collective-spin sensing, rather than a comprehensive comparison of experimental platforms or a practical guide to sensing experiments.