The United States has two major parties: Democrat and Republican. I won't go too deep into their political views here, but I'd like to bring to light for those who are curious, how their mascots came to be.
To do this, let's backtrack over a century. In the 1800s, the Democrats believed in small government, whereas the Republicans believed in big government. President Lincoln for instance, was a Republican. This change of party views shifted between the mid 1800s to the 1930s, when President F.D. Roosevelt was in office, as a Democrat.
Keeping this in mind, let's take a look at why the parties originally were represented by the jackass and the elephant. When Andrew Jackson (D) was running for President, his critics called him a "jackass." This name stuck with him while he was in office as well.
Putting Jackson aside, we have cartoonist, Thomas Nast (of "Harper's Weekly") to thank for the creation of both mascots. Nast, a Republican supporter (big government at the time) made both mascots famous in his 1874, "The Third-Term Panic" cartoon (Grant's third term).
Here, Nast depicted the Democrats (small government) as a jackass dressed in lions skin, scaring away all the other animals. The cartoon read, '"An Ass, having put on the Lion's skin, roamed about in the Forest, and amused himself by frightening all the foolish Animals he met with in his wanderings.'--Shakespeare or Bacon.'" In short, he wanted to alert the voters that though the jackass may seem like a lion, underneath it all, it's really, only a jackass, trying to fool everyone.
Today, Democrats say that the donkey is smart, brave and hardworking. The Republicans say that the elephant is strong, grand and dignified. I'll let you be the judge of that.
For your enjoyment, here are some campaign material depicting these two animals as representatives of the Democratic and Republican parties throughout the years.