What's the difference between hard, firm, and soft real-time?

They're distinguished by the consequence of missing a deadline. In a hard real-time system a missed deadline is a system failure, often catastrophic, think an airbag deployment, motor commutation, flight control surface, or a pacemaker, so you must guarantee the worst case is met, which requires WCET bounding and schedulability analysis. In a firm real-time system a result that arrives late is useless and is discarded, but a missed deadline isn't catastrophic and occasional misses are tolerable, for example a sensor reading that arrived too late to use in this control cycle is just dropped. In a soft real-time system a late result still has value, just reduced, and occasional misses degrade quality rather than causing failure, media playback, UI responsiveness, or telemetry, where a dropped frame or a slightly laggy update is acceptable. The engineering effort scales accordingly: hard real-time demands provable worst-case guarantees (analysis, bounded operations, margin), while soft real-time can be handled with measure-and-tune best effort. The classification tells you how much rigor a given deadline justifies, you don't spend hard-real-time effort on a soft deadline, and you must not treat a hard deadline as soft.