By default we split hit points equally in both pool. A character with 50 hit points has 25 luck and 25 health. This proportion may vary for classes or monsters.
When you take damage your Luck hit points are affected first then your Health hit points.
When you go to 0 Physical hit points you are dying. This occurs even if you still have Luck hit points.
Physical hit points are mainly healed by magic. Natural healing is possible but at a very slow rate or require special conditions like Trained healing skill. Magical healing affect Physical hits points first and then Luck hit points.
Luck hits point can be healed by various non-magical effects: Warlord feature, auto-healing, Short Rest, amateur healing, etc. Luck hits points are also healed faster in rest.
Some attacks may target only a specific pool of hits points. For example Necrotic spell can target directly Physical hit points. A Fear effect can target only Luck hit points.
I think I have seen this in an old ADnd unearthed arcana book..