The enemies in the game are called 'Heaven Smiles', and come in all different kinds, shapes and sizes. They're actually transparent and while in such a state, they can't take damage at all, so you have to scan the area to be able to see them. They laugh as they start to approach you, so it's easy to tell when they're nearby. The most important thing to remember is that Heaven Smiles detonate upon contact, most of which cause quite a bit of damage. So, they're pretty much walking mines - kill them before they get too close.
The appearance of the Heaven Smiles differ depending on the type, and their movements are pretty moderate paced. They also have a glowing spot on their body which is their weak spot. Shoot their weak spots to kill them instantly (critical hit kill). On a side note, each persona has their own catchphrase they say every time a Heaven Smile is killed by shooting at their critical spots, which is pretty cool.
As you progress through the game, you'll encounter different kinds of Heaven Smiles, and they will become more difficult to defeat.