An alternative definition: a real-time system is a system where the correctness of the computation depends on a deadline. For example, if I have a drone checking "with my current location + velocity will I crash into the wall in 5 seconds?", the answer will be worthless if the system responds 10 seconds later.

A real-time kernel is an operating system that makes it easier to build such systems. The main difference is that they offer lower latency than a usual OS for your one critical program. The OS will try to give that program as much priority as it wants (to the detriment of everything else) and immediately handle all signals ASAP (instead of coalescing/combining them to reduce overhead)

Linux has real-time priority scheduling as an optional feature. Lowering latency does not always result in reduced overhead or higher throughout. This allows system builders to design RT systems (such as audio processing systems, robots, drones, etc) to utilize these features without annoying the hell out of everyone else.