NPC damage types
There are four different damage types in EVE:

Electromagnetic (EM)
Kinetic
Thermal
Explosive
All weapons in EVE do at least one (often two) of these damage types, and every ship has different resistances against each. Understanding damage types helps you to survive longer and deal more damage to enemy ships:

If you know that your enemy uses weapons which deal a particular damage type, you can fit modules to your ship to increase your resistance to that damage type, making your enemy's weapons less effective against you.
Conversely, if you know that your enemy's ship has a low resistance to a particular damage type, you can use weapons which deal that specific damage type, making them more effective.
Native resistance Weapon damage
Shields Armor Lasers Hybrids Projectiles Disintegrators Drones Missiles
EM damage EM - - + + Yes (60%) Yes Yes (Amarr) Yes
Thermal damage Thermal - + Yes (40%) Yes (40%) Yes Yes (60%) Yes (Gallente) Yes
Kinetic damage Kinetic + - Yes (60%) Yes Yes (Caldari) Yes
Explosive damage Explosive + + - - Yes Yes (40%) Yes (Minmatar) Yes
Some points to note:

"Native resistance" is a tech 1 ship's resistance to damage before any modules or rigs are fitted. For instance, without any modules or rigs, shields are most susceptible to EM damage and most resistant to explosive damage. However, keep in mind that it's possible (with the right modules) to compensate for a ship's weaknesses; and tech 2 ships follow their own patterns of native resistance.
Thermal damage is not to be confused with heat damage, which is caused by overloading modules.
Energy turrets (often called "lasers") do about 60% EM and 40% thermal damage. The exact proportions depend on the frequency crystals used.
Hybrid weapons do about 60% kinetic and 40% thermal damage. The exact proportions depend on the ammunition type used.
Each type of projectile weapon ammunition does two (sometimes three) different damage types (one of which is always kinetic).
Entropic Disintegrators deal about 60% thermal and 40%