The Democratic party's self serving self appointed nickname is "America's Party".
I've heard that the democrats were called donkeys, and the republicans were called elephants. don't ask why, but I've always heard that in school. I've seen it in cartoons, too.
The long running nickname of the Republican Party is the Grand Old Party. To my knowledge the Democratic Party doesn't have a historic nickname like that. But what is interesting is that the "Democratic Party" was itself a nickname given to the party by its Federalist opponents. The word "democracy" has long been understood in the West to mean "mob rule" which is how Plato defined the concept in the Republic. Thomas Jefferson and other political opponents of the Federalists were called "Democratic" in the hopes of equating them with favoring anarchy and mob rule in the public eye.