Einstein (also known as “Einie”) was Dr. Emmett Brown’s favorite sheepdog in 1985, and the Brown family later adopted him. In 1985, his age was unknown, but he was fully developed and most likely “middle-aged” for a dog. He was the most recent of Doc’s beloved dogs to bear the name of a prominent scientist, in this case, Albert Einstein. As a result, it was only natural for him to become the world’s first time traveler.
Doc’s constant companion was Einstein. Despite employing the dog in many of his experiments, Doc cared greatly for his pet (apart from Marty and Jennifer, Einstein appears to have been Doc’s only friend). He would not jeopardize his friend’s safety. Doc created an automatic dog food feeder to make life easy for himself and Einstein. Einstein was also “mates” with Marty and was a cheerful, pleasant dog most of the time. As Doc mentioned in his letter to Marty, he only ate canned dog chow and was normally walked twice a day. Doc also gives him liver snacks, which he adores.
Einstein was in the Twin Pines Mall parking lot when Doc Brown performed his first time travel experiment. Einstein successfully rocketed forward in time by one minute, leaving at 1:20 a.m. and arriving at 1:21 a.m. on October 26, 1985, with Doc commanding the DeLorean time machine via remote, Einstein sitting inside it, and Marty filming. Einstein, being a dog, was “totally ignorant of anything that had occurred” and gladly returned to Doc’s truck, uninjured.
Minutes later, Einstein (who Doc had outfitted in a specially made radiation suit) barked from the truck, alerting Doc to the arrival of Libyan militants (from whom he obtained plutonium to power the time machine). After the terrorists murdered Doc and followed Marty around the parking lot until he vanished back to 1955, Einstein remained in the truck, most likely terrified of the gunshots.
Marty was able to warn the younger Doc in 1955 of his doom in 1985 in the past. Doc wore a bulletproof vest and survived the terrorists’ gunfire when 1985 rolled around again. Doc transported Einstein into the future after dropping Marty off at home. What happened to them while they were there is unknown, but Einstein was placed in a suspended animation kennel when Doc returned to 1985 to pick up Marty (and Jennifer, who happened to be there) and transport them to 2015.
Einstein stayed with Doc and Marty for the rest of their time in 2015 and hunted out Jennifer, who the police had taken to her future home. Doc still retained Jennifer’s pocketbook, from which Einstein got her fragrance. 
After completing their mission, the four returned to 1985, only to discover that it was 1985A – a nightmare other reality in which Biff Tannen was rich, powerful, and Marty’s stepfather. This was due to Biff’s 2015 self hijacking the time machine and gifting his 1955 self a sports almanac that contained the race results. Doc-A was committed to a mental facility in this world in 1983A, and it is unknown what happened to the alternate Einstein. His basket was discovered in the ruins of Doc-lab, proving that an alternate Einstein did exist at some point in this world.
Like Jennifer, Einstein (from the original timeline) was left behind when Doc and Marty returned to 1955 to make amends. When the chronology was reset to normal, it altered Einstein’s environment, albeit he would most likely be ignorant of this.
Doc could not return to 1985 after the DeLorean was struck by lightning in 1955. Doc told Marty, who was stranded in 1955, in a letter that he was quite happy living and working as a blacksmith in the Old West, and requested Marty to give Einstein a decent home whenever he returned to 1985. However, after seeing Doc’s tombstone in the Boot Hill Cemetery, Marty decided to go back to 1885 instead of 1985 to save Doc from imminent death at the hands of Buford Tannen.
Doc (who had spent ten years of his life in the late nineteenth century) returned to 1985 with his wife Clara and boys Jules and Verne. They adopted Einstein from Doc’s lab, and the dog became a family member. Even though it had been ten years since Doc had last seen Einstein, it had only been approximately ten hours since Einstein had last seen Doc. Einstein was with the family right up until the end when they left in the time machine for unknown times.