Doog, is a slang term for "Dog" that is primarily associated with pictures of a Pomeranian pup with a odd facial expression.
These photos may be photo-shopped to change the dog’s face and or to add accessory, such as hats, glasses, necklaces etc, they also consist of the words "Yiff Yiff" "arf" "woof" and "bark"

Roughly one to two years ago, a photo of Doog was posted on tumblr, in this photo i quote, "hey Shut the fuck up" ( Photo listed in gallery )
and shortly after someone decided to crop just the face of the dog and use it as a profile photo on Facebook, the trend has grown much more since then.

Sadly not much else is know since this is relatively new.