From Nookipedia, the Animal Crossing wiki
Grass is what covers the player's town (in the areas not covered by buildings, pavement, and dirt). In early December, snow falls and replaces all of the town's grass. On February 25th, the snow melts, and the grass comes back. In September the grass begins to become lighter to turn darker and browner in October and November.
Grass by game
In Animal Crossing, any part town not occupied by pavement or buildings is grass. There were only a few spots occupied by dirt, and those are designed in figure 8-like shapes.
In Animal Crossing: Wild World, more of the land was occupied by dirt rather than grass.
In Animal Crossing: City Folk, the player starts out with an almost complete town of grass. However, as the player plays the game, the grass starts to disappear. This process is called grass deterioration, and happens in areas where the player walks (and usually walks/runs most often).
Grass deterioration returns in Animal Crossing: New Leaf, but it is not as prominent as it is in City Folk. Also, the town is not filled with grass; there are patches of dirt scattered throughout the town.
There are six different designs for snow/grass in the Animal Crossing series. The patterns change when grass turns to snow and vice-versa. The patterns are as shown: