Much has been said about the questionably insane leadership practices (and life in general) of Roman Emperor Caligula, but arguably the most famous rumor is that he made his horse a member of the consul. It's hard to say for sure, as modern historians believe the historians of his time may have been trying to make Caligula seem even crazier than he really was, due to legends and the general nature of storytelling. Either way, Incitatus became the first horse to even consider the possibility of political power.
A small town in Washington called Milton was pranked by their own mayor in 1938. The Democratic mayor, Kenneth Simmons, sponsored and prepared for a Republican candidate for committeeman named Boston Curtis, who won the race. He claimed to have done so to prove the people supporting the Republican party "have no idea whom they support." The punch line of course is that Boston Curtis was a mule.
Bosco Ramos defeated two humans to become the Mayor of Sunol, California in 1981. He would remain mayor until his death, 13 years later. Also, he was a dog. The small town mayor simply didn't have many responsibilities, so it was possible for a dog to pull it off. This didn't stop Communist Chinese newspapers from using the story as a sign democracy didn't work, but maybe they were missing the point...democracy is darned cute sometimes.
Sometimes the population elects a dog because that dog had sound policies they could stand behind. Other times they elect a rhino to make a statement. Cacareco was a rhinoceros in São Paulo, Brazil who ran for city council and won in 1958. Students voted for the rhino in droves, feeling that he was just as good a candidate as any of the others, who were often corrupt and bound to lead dirty campaigns. Unfortunately, the corruption extended to the election process, and votes for Cacareco were discarded and a new election took place a week later.
Small towns in Alaska don't have much going on. Months of day or night followed by the opposite of that, the occasional dog sled race, but that's pretty much it. Which is how for 18-plus years the Mayor of Talkeetna, Alaska has been Stubbs the cat. He never raises taxes and the townspeople seem to love him, but the power has gone to his head a little bit. According to one aide, he will only drink water out of a wine glass. Classy.
Up and down politics, Americans have proven they are happy to elect anyone to any office as long as they seem like a swell person to share a beer with. Even if that person is a goat. The entire Clay Henry family of goats were local heroes in Texas, primarily because they didn't let the fact they were goats stop them from drinking beer. Clay Henry, Sr. was first elected Mayor of Lajitas, Texas in 1986, and his great-grandson holds the position to this day.
During the turmoil of the 1968 Democratic National Convention, the Yippie Movement, lead by Jerry Rubin and Abbie Hoffman, nominated a pig purchased by folk singer Phil Ochs to become the next President of the United States. His name was Pigasus the Immortal, but his story was short-lived. During his very first press conference, as Jerry Rubin demanded Pigasus receive Secret Service protection, all of the aforementioned men and pigs were arrested. Don't feel too bad for the pig, though...they say he squealed on his co-conspirators.