Crop produced by your farms is stored in the granary. The great granary offers you more space and keeps your crops drier and safer than the normal one.
After building up one great granary to level 20, additional great granaries can be built.
The level dependent attributes (such as construction costs, generated culture points and capacity) can be found here.
For the full table of construction times click this link for server with 1x speed and this link for server with 3x speed.