Einstein (often known as "Einie") was Dr. Emmett Brown's pet sheepdog in 1985, and later pet to the Brown family. His age in 1985 was unknown, although he was fully grown and most likely "middle aged" for a dog. He was the latest of many Doc's pet dogs named after famous scientists, in this case Albert Einstein. It was, therefore, quite fitting that he became the world's first time traveler.
Einstein was Doc's constant companion, and despite being using the dog in many of his experiments, Doc did care about his pet deeply (since apart from Marty McFly and Jennifer Parker, Einstein appears to have been Doc's only friend), and he would not compromise his friend's safety. Doc built an automatic dog food feeder to make both his and Einstein's lives easier. Einstein was also "friends" with Marty and was usually a cheerful, friendly dog. He only liked canned dog food, as Doc pointed out in his letter to Marty, and was usually walked twice a day. He also enjoys liver treats given to him by Doc.
Back To The Future
Einstein was present at the Twin Pines Mall parking lot, where Doc Brown conducted his first time travel experiment. With Doc controlling the DeLorean time machine via remote, Einstein sitting inside it and Marty filming, Einstein successfully blasted forward in time by one minute, leaving at 1:20 am and arriving at 1:21 am on October 26, 1985. Einstein, being a dog, was "completely unaware that anything happened" and happily went back into Doc's truck, unharmed.
Minutes later, Einstein (whom Doc had put in a specially-designed radiation suit) barked from the truck to warn Doc that the Libyan terrorists (from whom he obtained plutonium to power the time machine) had arrived at the Mall. After the terrorists murdered Doc, and chased Marty around the parking lot until he disappeared back to 1955, Einstein stayed in the truck, probably scared of the gunfire.
When in the past, Marty was able to warn the younger Doc in 1955 of his fate in 1985. When 1985 rolled around again, Doc wore a bulletproof vest and survived the terrorists. After dropping Marty back home, Doc took Einstein with him into the future. What happened while they were there is partly unknown, but Einstein was left in a suspended animation kennel when Doc headed back to 1985 to pick up Marty (and Jennifer, who happened to be there).