There are two basic types of attacks. Specific behaviors vary from character to character, and these moves may involve a single blow or a flurry of blows in a combo. In the case of combos, generally thefinal hit will count for more, so interrupted combos are not very effective. Chaining multiple moves together in unblockable form is generally impossible, so this is a game of careful, considered strikes not twitchy button mashing.
The basic combat comes down to a balance between two main types of attacks: HP and Bravery. Each of these attack types have completely separate moves that can be assigned from a larger move set. Three ground attacks and three (different) aerial attacks are available for each character.
Bravery is effectively a damage multiplier. You need it in order to deal damage, and if you can get it high enough, you can often end a fight with a single HP Attack. Bravery attacks are usually faster, more effective, and more combo-intensive, although this varies from character to character.
Attacking an opponent with a Bravery attack not only robs that character of their Bravery, but adds to yours. Bravery is much easier to deplete than HP, but deals no permanent damage. If you push your opponent's Bravery down to zero, you will earn a "break," which will greatly amplify your Bravery, and leave your opponent temporarily without any means of dealing HP damage to you. This is the fastest way to end most fights.
Because this is a game of fewer blows, it's a much more defensive game than most fighters. You have a few means of doing this, and you'll need all of them. Blocking is the most obvious. If you time your block just before a blow lands, you'll also throw an enemy off balance, allowing you a brief moment to counterattack with impunity. Blocking only stops attacks from the front, however, and impedes movement.
In addition, certain characters have attacks capable of crushing your guard. Against long range characters, the most effective defense is simply evasion. Learning to read characters' moves and animaions is an important skill, and after a while you'll know when to get away. In addition, many moves leave your opponent vulnerable for a brief moment. Watch Onion Knight's strikes for very obvious examples. Once you get better at those two techniques, you can become skillful at dodging. To dodge, simply perform a dash right before a blow lands. This will move you out of the way, but only by a short distance, possibly still in range to strike back.
Certain moves will knock the victim into or through the air, triggering the aggressor to chase them and engage in a mid-air clash. This is a simple slow-motion game of timing. When it's your opponent's turn to attack, dodge at the right time, and you'll earn a chance to strike back. These can often go back and forth and be fruitless, but learning to read moves will help your odds tremendously.
The third meter you need to be aware of is the EX Guage. This vertical bar is similar to a "super meter" in other fighting games. As you clash with your opponent, you'll notice small orbs of light appear. Passing through these increases your EX a tiny amount. EX Cores (which look like glowing bells) appear. These fill your EX meter by much larger increments.
Once your bar is filled, it will turn yellow. Holding R and tapping Square will send you into EX Mode, and the bar will slowly deplete. There are different perks to EX Mode, which you can select outside of battle, that include stat boosts and regenerating health. In addition, the moment you engage EX Mode is an opportunity, as it will break any combos that your opponent may be inflicting on you.
Most importantly, EX mode allows you to initiate a Burst Attack. This is done by landing an HP Attack (if this attack is a combo, you must land the final blow). A prompt will appear, and you can press the Square button to engage. This will initiate a series of blows, and will involve some sort of a minigame. This is usually some variation on Simon Says or mashing button(s), but some of them are more unique. Your success in this (which is sometimes all or nothing and sometimes has degrees) will determine the effectiveness of the final blow.