The elephant and the donkey as symbols for the Republican and Democratic party were popularized by the political cartoons of Thomas Nast in the late 19th century. Nast is probably the most influential political cartoonist, making use of visual symbols to represent groups and ideas (pretty necessary when your art form is primarily visual).

You can find this in almost any textbook on U.S. history, but I don't have one in front of me. I did find a journal reference for this, which also goes into some interesting analysis of Nash's influence on 19th century politics.

edit: You may need JSTOR access for this article.

The donkey came about because of the first recognized Democratic president, Andrew Jackson. At a time when mudslinging was relegated to papers and posters, Jackson's opponents referred to him as a Jackass (which he definitely was), meaning a "stubborn and stupid animal." The Democrats kept the image because of its common-man lingo, and it's been used ever since, despite not being their official logo.