Two of the most distinct aspects of the Persona 3 battle system are the knocked-down condition and the "1 More" effect. Every time either an ally or an enemy hits a target's weakness or hits with a critical hit, the enemy will be knocked to the ground and the attacker can perform an additional action. Further, if an ally or enemy makes a melee attack and misses, there is a chance that character will stumble and fall to the ground. Finally, if all enemies are knocked down, the team can perform a free "All-Out Attack" on every enemy target.
The knockdown/"1 More" effect of hitting an enemy's weakness is incredibly influential to gameplay. With this system, so long as you have the attacks that let you attack the enemy's weakness, you can attack until every enemy is knocked down and an "All-Out Attack" becomes available, which is usually enough to defeat any normal enemy group. This means that, so long as you are prepared, you can defeat foes quickly and easily, making long trips into Tartarus a lot more manageable. Of course, the controllable characters also have weaknesses, so the enemy can easily take advantage of a player's mistake and turn even an ordinary battle into a desperate fight to survive. Because of this, the game strongly rewards good planning and fighting with intelligence, rather than brute force or simple level grinding, which is something I always like to see in games.
One particularly important part of this whole system is the fact that it is nearly impossible to be 100% sure you can take advantage of a foe's weakness. If every allied character had the ability to use every kind of elemental attack, then the entire system would fall apart. In such a situation, every time a character's turn came up they would be able to knock down every foe and launch an All-Out Attack, and the game would be far too easy. However, Persona 3 prevents this by restricting every character down other than the hero down to usually a single weapon type and a single magic element or something equivalent, and limiting how many times the hero can change Persona in one turn. These restriction mean that it is impossible that every character can attack the same vulnerability, and that it is difficult to knock down a mixed group of foes. These restriction mean that even such a powerful game mechanic is not problematic, and they also add to the importance of planning and strategy in the game.
Another major feature of this system is that it makes getting the first turn in battle incredibly important. If you can sneak up on a Shadow you can easily defeat it, but if you are ambushed the enemy may knock down some of your characters and inflict a lot of damage. Thus, the game encourages you to get good at preemptively striking the Shadows in Tartarus and avoiding their ambushes.
Something I particularly like about the system is that an All-Out Attack is not always the best choice. All-Out Attacks are certainly powerful and useful, but they do have drawbacks. Whenever you use an All-Out Attack you have to give up a "1 More" action and all enemies who have been knocked down are able to get back to their feet. You do a lot of damage, but you give up an option and the enemies can launch counter-attacks without interruption. Because of this trade-off, there are many situations where it is a better idea to cancel the All-Out Attack and leave the enemies knocked down, particularly in situations where the enemies have powerful attacks and the heroes are injured. While All-Out Attacks are a useful way to quickly defeat weak foes, they are not a guaranteed way to win every fight. Instead, they merely serving as an additional option in battle, adding to the already impressive tactical complexity of this game.
Needless to say, I really like the interesting variety and solid challenge presented by Persona 3's combat system. It has the kind of complexity and enjoyability of combat that I usually have to turn to Strategic RPGs or pure Strategy games in order to find, and I like seeing that in traditional RPGs (though I suppose it is inaccurate to call Persona 3 traditional).