there are 7 different combos: saucy, dandy, fierce, one that i forget the name of, bellator, maquina, and sanctus.
combos can be initiated by certain cards that match the criteria. the higher the combo level of a card (max 200) the higher chance the combo will activate.
Dandy, fierce, and the one i forget because i rarely use gun cards:
these are the combos related to card icon types. the staff icon can activate dandy, even pixies can activate it, though their combo levels are generally so low that they won't affect it. the sword icon activates fierce, and the gun icon activates the one i don't remember.
These combos are activated by having two or more of those card types, and can activate multiple times on different cards, though not multiple times on a single card. These seem to boost atk/def by roughly 5%
bellator, maquina, and sanctus combos:
these combos are activated by having two or more of the same nation's card in the battle. I have not seen an unafiliated combo come up yet. These tend to activate easier than the previous combos, but have far less impact on a card's stats, something along the lines of 2-3% boosts.
Finally, there is Saucy combo:
This combo only seems to activate rarely on high combo level cards. It activates on all pirate types from what i've seen, as long as their combo is high enough. I most see it when calling in my guild master's athena as it is maxed combo level. This combo gives a huge boost, though i can't pinpoint the actual value. it lies somewhere in the 10-20% range, and can drastically change a fight.
I hope that helps!
Edit: Forgot to mention, combo level also affects the rate a skill activates, and the chance of it affecting another card. (i.e. athena's leviathan activates, yet a combo level 20 akira doesn't get the boost from the skill as it's combo level was so low that it has a rare chance of being affected.)