The basic requirement for a real-time OS (RTOS) is that it can guarantee that tasks will be executed and finished within a certain time span. This allows one to use them for real-time applications where variability (jitter) between the execution times of a batch of the same task is not acceptable.
From this, we can draw the basic distinction between hard and soft real-time OSes: with low jitter, the OS is hard real-time, as it can guarantee that a given task will always be executed with practically the same delay. With higher jitter, the OS can usually but not always execute a task with the same delay.
Within these two categories, we can again distinguish between event-driven and time-sharing schedulers. The former switches tasks based on priority (priority scheduling), whereas the latter uses a timer to regularly switch tasks. Which design is better depends on...
