What is the origin of the donkey and elephant as the symbols of the Democratic and Republican parties?

There was a cartoon made by Thomas Nast who made the symbol as an elephant as Ulysses S. Grant was running for a third term and he made a cartoon to show that the elephant was unmovable when calm, but destructible when excited. The donkey came from when Andrew Jackson was in office, they made a donkey to represent him because they thought of him as a "jackass".

Presidential candidate Andrew Jackson was the first democrat to be associated with the donkey in 1828. His Republican opponents thought his views made him a "jackass". His campaign slogan was, "Let the people rule". They though it was such a ridiculous idea then and they still think it's a ridiculous idea! Jackson liked the analogy and used it to his advantage in his campaign.

The cartoonist Thomas Nast was a popular political cartoonist who used several animals to represent the political parties. He is credited with popularizing the donkey and the elephant having settled on those particular two animals by 1880.