Hit points, often abbreviated to HP, are used in many games to determine, how much damage an object within the game (e.g., a character, unit, or building) can withstand before it is rendered unusable.
In Rise of Nations, all units and buildings have a certain amount of hit points. If a unit or building's hit points drop to 0, it is permanently destroyed. The only exception to this are besieged Cities, which after losing all of their hit points, come into the posession of the first army to get close enough to them, and automatically begin to restore their hit points as soon as this happens. The only way to truly destroy a city is by razing, as long as it's not a Capital City.
A unit's or building's hit points will drop if it is being attacked by another unit or taking attrition damage from being in enemy territory. A building's hit points can be restored by ordering Citizens to repair it. Units will heal lost hit points while they are garrisoned in a building, by a specific Patriot, or a French Supply Wagon.
Generally, hit points of a building determine the time needed for it to be razed.