The combat system is pretty simple, though fun. Kind of reminds me of Shinobi, Otogi and Bujingai. The animations are a bit stiff and the camera goes into these cinematic angles at times that can become disorientating, but it's not too bad once you get used to it.
So, this is the first boss in the game. The flashy air combo is this game's main feature and so I'll explain how it works. It's basically a counter attack.
When locked on, wait for the ring to turn red, which only happens during specific points in an enemy's attack or recovery. Once it turns red, connect an attack, and the character launches the targeted enemy along with surrounding enemies. From here you attack and then dash to the next enemy.
The rules are weird though. It's not as free as Shinobi for example, where you can do something else in the air, or just dash at enemies whenever you want. In this game, there is a limit to these air attacks. In the bottom right you'll see a circle of blue orbs. How many of these you have full determines how long you can stay in the air attacking. A timer basically. They fade out the longer you continue your air attacks. You replenish them by attacking enemies with regular attacks.
Ok, so once in the air, you press square to attack the targeted enemy, then you press triangle, which is a long range dash, to move to the next enemy. You must hold down the lock on button the whole time so it targets the next closest enemy.
There is a rule here though, that I don't really like and that's that each sword has an air combo that HAS to play out, so in this video it's the starting sword. After you launch, its air combo has 3 slashes. You CANNOT dash to the next enemy until you have done all three slashes. Even if you kill the enemy in the first slash, you can't just dash to the next, you have to slash nothing two more times. The upside is that these empty slashes move you towards the next closest enemy, then you can dash to close any remaining distance. If you try to dash after the first or second slash, you just drop and the sequence ends.
For bosses, single targets with high HP, you can just do like in this video and combo them until your air time meter runs out. You still need to dash after each combo though.
So the command would look like S,S,S,T(DASH),S,S,S,T(DASH) repeatedly. The timing isn't difficult but if you fail to dash or attack fast enough you'll drop. Different swords have varying attack speeds, number of hits, and animations.
There are also magic attacks, charge attacks, and items that boost your stats in battle, as well as one that works like Shinobi's shurikens, allowing you to paralyze and launch without waiting for the counter window.