A Burrow is a naturally occurring object from the Reign of Giants DLC that randomly spawns in Grassland Biomes. Moleworms will spawn at Dusk and Night from Burrows and start roaming around the area, looking for minerals and other ground-based items to store inside the Burrows.
Burrows can be dug up with a Shovel to randomly yield Rocks, Flint, Nitre, and Gold Nuggets. When dug, Burrows will also drop any item stolen by Moleworms as long as there weren't more than 50 of them.
If a Burrow is dug up, the Moleworm that inhabited it will make a new one after some time. This makes Burrows renewable as long as the Moleworm is not also killed. If a Moleworm is killed, the Burrow will spawn a new Moleworm, in the same way Rabbit Holes spawn Rabbits. However, it is worth noting that even if all of the Moleworms and Burrows in a world are killed or destroyed, respectively, they are still renewable through both Catcoons and Tumbleweeds. Because Burrows drop minerals and Moleworms can create new Burrows, this makes Rocks, Flint, Nitre, and Gold renewable.