The donkey is the democratic party's symbol. 1828 the candidate Andrew Jackson has been called "jackass" by his opponents and in 1837 the first cartoon with a donkey was published. When that symbol was published in a political magazine in 1870 the donkey as the democrats symbol was finally established.
The logo of the republic party is the elephant. It all started with a cartoon and a magazine publication, too.
For the republicans the elephant stands for power and intelligence.