A Tower is present in every lane in Artifact. They are the main objective of the game, and so every strategy has to have a plan for protecting your towers and destroying those of the enemy. After you destroyed a tower, the Ancient will be exposed on that lane. Destroying two towers or the Ancient will win you the game.
Every Tower starts with 40 health. To attack it, a unit must simply not have an attack target that round, which is easier said than done. Other ways to damage Towers include Siege Damage and Certain specific cards.
Artifact 2.0 Beta
In the beta preview, it is said that towers have 30 Health, but the value can change depending on the gamemode.
The Ancient acts exactly like a tower, but has twice the health, at 80.
Excess Tower damage (Overkill damage) does not carry over to the Ancient.