The granary is where you store your Grain. Raising it's level increases the amount of grain you can hold.
This structure is of the lowest priority for leveling. The Granary is a potential target when a player destroys a building (-1 level) in PvP Combat.
Starting at 2,000 capacity, levels 2 & 3 increase your grain capacity by 500 per level, 4-15 increase your capacity by 1,000 per level. Each level afterwards increases the capacity by 5,000 per level.