Dog is one of the main characters introduced in Half-Life 2 and is described by Alyx Vance as being built by her father, Eli Vance, to protect her after the events at Black Mesa. Alyx has been "adding to him ever since" however and he is now bigger than any adult.
Your first meeting with Dog takes place at Black Mesa East when Alyx takes you outside and teaches you how to use the Gravity Gun. The training session plays out as an elaborate game of catch with Dog throwing objects for you to catch and toss back to him. The game is interrupted however when Scanners fly over and the area is blanketed with Headcrab Canisters. Dog helps you escape to Ravenholm.
He provides further assistance later in the game, during a fight through the streets of City 17 where he throws cars at Combine soldiers before finally being carried away on the back of a Dropship, and nearer the end where he finds a way for you to sneak in to the Combine Citadel.
It is unclear what happens to Dog after the destruction of the citadel, but a teaser trailer for expansion pack Aftermath implies that he will return.