The games that usually have dogs in them fall under two main categories. Either action-adventure games that use an NPC dog as a gameplay device, or simulation-oriented games where players interact with a dog.
Dogs as an ally
Dogs are sometimes used as an ally to aid the player in combat, or to assist in the search for collectibles and/or treasure.
One of the most well known allied dog is Shadow from the Dead to Rights series. In the original Dead to Rights Shadow could be used like a weapon, instantly killing and collecting the weapons of an enemy to give to Jack (the protagonist). At a later stage the player took control of Shadow to track down bombs hidden in a hotel. In the remake Dead to Rights Retribution, shadow became AI controlled and would attack any enemy near Jack, and would occasionally be controlled by the player for brief stealth sections, whereupon we discovered the little known fact that dogs can see through walls.
The Heroes of Fable 2 and 3 both have faithful companions, these dogs will assist in combat and point players in the direction of treasure. They can be named, their breed changed (mainly through DLC) as well as learning tricks, and upgrading their stats, with books.
Dogs as enemy type
There are also various games featuring dogs as a type of enemy the player has to fight. The best and most well known example for this would be the Zombie Dogs from the Resident Evil series.
The Call of Duty series is also known for it's vicious dogs, as they included dogs in every game since Call of Duty 4 . They are widely used in every game and mode. The Treyarch games feature dogs as a Killstreak in Multiplayer Mode with which the player can call in dogs that will attack the opposing team. More obscurely, Zombie Mode of World at War and Black Ops features dogs known as Hellhounds, which are supposedly zombified nazi dogs. In the Modern Warfare games by Infinity Ward, dogs are mostly used in the single player campaign, prominently featured 7 times in the campaign of the first game. Dogs also appear in some levels of Spec Ops mode and the Survival mode new to Modern Warfare 3. Notably, there are dogs with C4 strapped to them attacking the player in MW3.
In 2008, two of the blockbuster games featured dogs as NPC companions, Fable 2 and Fallout 3. While Fallouts Australian Cattle Dog Dogmeat is only one of many available companions and is not linked to any questline, he was the only companion featured in various trailers and screenshots prior to the release of the game. The dog in Fable 2 on the other hand is an integral part of the gameplay and story, as the player meets the dog very early in the game and is not able to part ways with it. The dogs (as well as the players) appearance heavily relies on the players actions, making the dog look similar to a Golden Retriever if the player is of a good alignment as opposed to a Black Labrador, if the player is evil. The DLC See the Future allows the player to change the dogs breed to Dalmatian, Bloodhound or Husky.
Gameplay-wise, the dog can be used to find and dig up treasures and fight enemies. It's also possible to teach the dog tricks to either impress or frighten other NPCs. The dog has no name, but can be given one by equipping it with a dog collar. Peter Molyneux's goal with the dog was to give the player a companion to get emotionally attached to.