In Animal Crossing: New Leaf, grass can be worn away if you walk over it frequently. If you take the same path when you walk and run around your town, you will eventually create dirt paths in those areas. This page has more details about grass deterioration and growth.
Grass will wear away if you walk over it frequently. The rate of grass deterioration is unknown, but it wears away much more slowly than in Animal Crossing: City Folk. In City Folk, grass paths would start to show after a week. It takes much longer in New Leaf.
Grass gradually grows back. Grass will grow more quickly if there are lots of trees and flowers in the area. If you plant flowers on a dirt patch, where there is no grass, be sure to water those flowers. This will encourage the grass to grow back. Rain and snow also help the grass to grow back.
If you put patterns on the ground in your town, the grass under those patterns will not wear away if you walk on them. To put a pattern on the ground, simply open up your patterns by tapping the pencil icon at the top of the touch screen, then tap a pattern and choose to drop it on the ground.