Damage refers to when weapons reduce a unit's (or shield's) hit points. If a unit's hit points are reduced below zero, the unit is destroyed. Similarly, when a shield's hit points are reduced below zero, the shield collapses and must recharge. A unit with zero health will not neccesarily die, but will continue to live until its health is brought down below zero, although it is very rare to see a unit with zero health.
Different weapons have different ways to deal damage. Normal weapons only deal damage to a single target, whereas weapons with splash damage (e.g. artillery, nukes, and bombs) deal damage to multiple units at the same time within its area of effect. Other weapons, deal their damage over time, along with an area of effect. Any units that moves into the area while the damage is still being dealt will take damage until they either leave or the effect's time expires. Another type of damage, overcharge damage, does massive damage to normal units but next to no damage to command units. The damage created by the death nuke of an ACU is also unique in that it does varying damage to different units, dealing the most damage against units and the least against buildings.