When you're in the field, you press a button to attack, and another button to activate turn-based battles. The idea is that you can get in a few solid hits or stun an enemy in "action combat" before going into battle for real-real, or that you can wipe out smaller monsters that are well below your level in a few hits. If you want, you can activate turn-based battle after a single hit to an enemy. Bosses and other "big" enemies, you get launched directly into turn-based mode.
It's a pretty cool system, and you'll definitely be doing most battles turn-based. Turn based combat itself is literally old-school turn based.
On the left-hand side of the battle screen, you see the turn order from top to bottom. Different turn positions have bonuses (the blue and green hearts, in this case) which indicate that when that character's turn arrives, they'll heal some EP (used for casting spells) or CP (used for casting super moves). The second blue heart bonus is going to a monster here, so one strategy would be to try to delay that monster's turn so that the EP bonus for that turn slot gets pushed on to one of your own characters for maximum benefit. The bonuses are randomly generated but are used to keep fights fresh and interesting to avoid the "press X to win" that most JRPGs eventually get to.
As others mentioned above, some fights (mostly boss fights) can be absolutely brutally difficult if you don't actually take advantage of your character's move sets or control the battle turn order. You're free to position your characters (when it's their turn) because most attacks need to be lined up correctly to work optimally. Some attacks go in a straight line, some a circle around the character, some in a cone in front, etc.
I hear people complain a lot of times that they wished JRPGs just kept refining turn based combat after Final Fantasy X. If you've thought that before, this game is here to restore your faith in the genre.