This is a quick overview of the different kinds of damage that can happen to you (and that you can do to others!). You can see your character's stats for all of the different things described here in-game on your Persona window, under the Stats tab.
Direct vs. Indirect Damage
When you attack someone, or they attack you, and the attack deals, say, 10 damage, that's "direct" damage. If the attack causes lingering damage, that's called "indirect" damage. For instance, if you shoot somebody with a fireball, the immediate attack will do Direct Fire damage, and if the fireball sets them on fire, they'll take Indirect Fire Damage for as long as they're ablaze. Rule of thumb: if it lingers, it's Indirect.
This distinction is sometimes important because some equipment, abilities, and curses alter only a certain kind of Direct or Indirect damage.
There are 12 different damage types. They can be lumped into four categories: Physical Damage, Elemental Damage, Body Damage, and Esoteric Damage. (The categories aren't really important. Sometimes you'll get enchanted items that improve all of one category, but that's pretty rare. The categories are mainly just a helpful way to think about them.)
Slashing: a physical damage type. Razor-sharp claws, swords, most any kind of blade deals Slashing damage. Slashing only comes in Direct damage; there's no such thing as Indirect Slashing damage.
Piercing: a physical damage type. Spears, arrows, rapiers, even especially-pointy teeth deal Piercing damage. Piercing only comes in Direct damage; there's no such thing as Indirect Piercing damage.
Crushing: a physical damage type. Clubs, fists, crushing jaws, and hurled boulders all deal Crushing damage. Crushing only comes in Direct damage; there's no such thing as Indirect Crushing damage.
Fire: an elemental damage type. Intense heat, dragon breath, and, well, any kind of fire deals Fire damage. Indirect Fire damage can be put out by jumping in water.
Cold: an elemental damage type. Icy water, snowstorms, and magically-forming ice blocks all deal Cold damage. The Kur Mountains are infamous for dealing lots of Indirect Cold damage -- the biting wind damages you just by standing in it! Fire Mages are usually adept at manipulating both Fire and Cold, because they use very similar temperature-controlling magic.
Electricity: an elemental damage type. Lightning, sparks, and shocks of any kind deal Electricity damage. There's no such thing as Indirect Electricity damage. (At least not right now... it hasn't come up!) You can't safely deal Electricity damage while underwater. (Unless you're an eel.)
Poison: a body-damage type. Injected venom, ingested poison, and poisonous gas clouds all deal Poison damage. Some things deal Direct Poison damage (like when you get hit with a big ball of poison in the face), but most Poison damage is Indirect, a slow lingering pain. Long-lasting Indirect Poison damage can be cured with First Aid and certain other skills.
Trauma: a body-damage type. Broken bones, blood loss, and having your spleen teleported into space are all examples of Trauma damage. Almost all Trauma damage is Indirect, causing ongoing debilitation. Lots of martial arts moves deal Direct Crushing damage at first, then deal lingering Trauma damage (due to broken bones or damaged organs). Battle Chemists can learn a mutagen formula that makes them (and their allies) immune to Trauma damage.
Psychic: a body-damage type. Mind probes, psychic blasts, and any kind of brain-affecting attacks are Psychic damage. Most psychic damage is Direct, like when a mentalist shoots you with a brain beam. But some Psychic damage is Indirect; for example, a common headache deals small amounts of Indirect Psychic damage for as long as it lasts.
Acid: an esoteric damage type. Corrosive liquids and gasses deal Acid damage. Acid Damage tends to rapidly eat away at your Armor, but some Acid Damage can also eat away at your flesh just fine. (This damage type encompasses various ways your body and armor can be corroded, not all of which are strictly acidic! For instance, being splashed with bleach might deal Acid Damage, even though bleach is a base, not an acid.)
Darkness: an esoteric damage type. Certain nature gods have infused the world with a special kind of magic, which can be used to empower the dead and make them Undead. That same energy can be turned into attacks which deal Darkness damage. Anything that instantly rots your flesh, rips at your soul, or tries to pull you bodily into the nether realm deals Darkness damage. Undead are all completely immune to Darkness damage.
Nature: an esoteric damage type. Swarms of stinging bugs, toxic spore clouds, fungal infections, and clinging leeches all deal Nature damage. Things that deal Nature damage tend to be hard to kill with physical attacks -- for instance, it's hard to Pierce a swarm of bees to death. Elemental damage tends to wipe them out fast, though!