Imagine you are sitting on a spinning chair and you start to spin around really fast with your arms stretched out. You can feel something pulling on your hands, making it harder for you to keep your arms out straight. This is because of something called angular momentum.
Now imagine if you were spinning around as fast as a spaceship traveling near the speed of light. Because you are going so fast, weird things start happening. One of those weird things is that time slows down for you compared to someone who is not moving as fast. This is called relativity.
But even though time is different for you, your angular momentum stays the same. This means that the force pulling on your hands stays the same even though time is different. So, your angular momentum is "relativistic".
Basically, relativistic angular momentum is just like regular angular momentum, but it takes into account the weird things that happen when you are moving really, really fast.