There are a lot of weapons in War Robots but all of them can be divided into 4 big groups based on their damage type.
Kinetic damage is inflicted by solid projectiles that resemble bullets. Weapons of kinetic type deal greater damage to physical shields (but don't bypass those completely) and are less effective against energy shields. In fact, energy shields block kinetic damage but can be relatively quickly destroyed by kinetic weapons.
Energy damage is inflicted by energy shells. They are able to bypass standard energy shields (Ancile) but can be blocked by physical shields, although they still damage those. Flamethrowers are also a subtype of energy weapons.
Explosive damage is inflicted by different types of explosive rockets/missiles. All rockets can deal area damage while homing missiles lack this ability.
Sonic damage is inflicted by directional sound waves. This type of damage can't be repaired in the field but can be blocked by energy shields.
Damage over time (DoT) is a special effect and a subtype of damage. If the effect is applied it deals damage to your target even when you stop shooting until the effect wears off.