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Real time OS is meant to execute the real time applications. ex. Software and system in aircraft, robotic software etc. In simple words the action should happen is specified time after the trigger. In simple words, In Home PCs if program takes more time or less time doesnt matter. We will wait till its completion. But in real time if an application is expected to complete in 4 secs then it must with accurate results. Ex. Movement of a Robot Hand.
What topatil said, basically when you run commands or execute functions they have a time limit in which to return an answer. In most applications that time limit will be very short, other applications the limit is larger. Real-time OS's are generally used when you can't have the system getting bogged down in any one part because other threads need to run and are time critical, ie they must respond/be active within a certain time span.
In any real-time system, failure to complete an operation within a certain period of time can cause catastrophic failure of the entire system. I just thought I'd add that, though the above two answers are far better.