The special symbol of the Nazi Party under Hitler was the swastika or, as it was called in German, the hakenkreuz (which means "hooked cross"). This symbol became the symbol of the party and was used on their party's flag.
The swastika was used because it went well with the Nazi ideology that said that "Aryans" had created all civilizations. The swastika had been found, for example, in ancient Indian culture and in the ruins of Troy. Germans who believed in this sort of thing decided that the swastika was a symbol associated with the Aryans who had created civilization and who had ended up in Germany.