The Piglin Royal will be fought in a special throne room (don't mind the throne, its new textures are almost done so it hasn't been imported into the mod yet). This structure is pretty rare and can often spawn hidden, but the boss bar will give away that its nearby, so you'll just need to look.
It will be your job to get close between summoned waves as it is fully immune to ranged damage.
At 70% and 30% health, it will spawn a special shielding ring that makes it immune to all forms of damage, as well as two piglin brutes to enforce its decree, so it will be your job to take out the new targets thrown at you.
The Royal will also give you a new effect called Piglin's Decree, which will make you unable to break/place blocks, so you will need to fight it on its terms. This effect will obviously disappear when you take it out.