Combos work in multiple parts in this game.
First you have the normal attack, each character will attack once a turn and use a normal attack the will hit the enemy a certain number of times which varies with each ninja. The normal attack may have a chance to activate a special version that can inflict status effects. For example Sasuke's normal attack is a 1 hit attack but can proc a 3 hit attack instead which will effect the enemy with the float status.
Status effects can last anywhere from just that attack to multiple rounds later and include float, poison, burning, repel and others.
Certain ninjas have passive abilities that let them inflict or chain off of different status effects. As mentioned above Sasuke can give enemies the float status, be he also has a passive that lets him attack enemies with the float status and cause the repel status. So Sasuke can cause float through his normal attack proc'ing the improved version and then his passive will proc, removing the float status and replacing it with repel which causes damage to that enemy and all enemies in the same row behind it.
Different ninjas can cause different status effects and have different passive attacks that chain with different effects so by using the right formation you ca increase the number of attacks that you can do per round.
Some ninjas also have abilities that will proc after a certain number of consecutive hits and will then deal extra hits, Sasuke can launch fire* at 2 enemies if you get a chain of 10 or more.
Passives may only be used once per round so you can't have Sasuke's improved attack cause float to chain into his repel and then have Naruto's improved attack cause float so Sasuke can repel again, the passive repel effect will only proc once per round.
In addition to all of that every ninja will have a special skill that can be used that usually costs chakra(Lee's skill, as well as a few others, does not cost anything). These special skills can do a wide variety of things from poison the enemy team to paralyzing them or you can heal your own team. All skills in this category have varied cooldowns and cost various amounts of chakra. You start the first round with 20 chakra and gain an additional 20 each round.