On one hand, you have the physical fighting itself. Traditionally, is was the weak point of western action movies for a long time. If you watch most of Chuck Norris fight scenes, the choreography and the physical ability of the performers is terrible. This has been getting better since the rise of Jackie Chan and other Hong Kong Martial Arts films, but is far from over (looking at you, first season of Agents of Shield). Some of it has to do with the general difference in weight, most chinese stunt men are simply in another class of athleticism and speed than western action heroes that generally fall into the (light) heavyweight category. More on this problem later.
(Lady Sif feels the pain of bad choreography)
The other big aspect of filming a fight is the choreography. All the athleticism of the fighters is wasted if the scene is filmed in a boring or unaccentuated way. Classic examples of this are bad Shaw Bros movies that just film the insane (and insanely silly) fights mostly from fixed total with both fighters in plain view (think Beat-em-up video game perspective.)
(round 2 ... FIGHT!)
Although there is much to talk about on the subject, some great aspects are captured in this Youtube video by the channel "Every frame a painting". It focuses on comedy, but also has quite a bit to say about action scenes.