A God's statue is a statue of one of the gods of Tyria:
These statues, while being places of worship for the inhabitants of most of Tyria, are not merely ornamental, but when the World has the Favor of the Gods, by kneeling in front of a statue, players can summon an avatar of the respective god and interact with it.
In Cantha and Elona, each Resurrection Shrine of the explorable areas includes a particular statue, however Resurrection Shrines in the Realm of Torment have no statues. The PvP Resurrection Shrines do not have statues.
In Eye of the North expansion, there are no god's statues.
Note: While exploring, players will find murals of the five gods. While beautiful, they are different from statues.
Effect of Favor
Depending on whether the world has the Favor of the Gods the various statues of the gods scattered across the game world will change in their outward appearance to reflect this. If the world holds the favor then the statues become active, allowing you to interact with their avatar. Otherwise they are inactive and useless. In addition, if the world gains or loses favor while you are in a zone containing a statue, the statue shifts from "inactive" to "active" (or vice-versa) with an animation. For example, if the world gains favor, collapsed statues of Dwayna will be struck by lightning and the pieces will float into place.
If a statue is "active" (this means if the world has the Favor of the Gods) you can summon the god's avatar at their statue to interact with it. You do this by walking up to the inscription at the statue base and typing "/kneel" in console.
For details on appearance and locations of the statues, and possible interactions for each of the avatars, check their respective statue articles.