Former poli sci student here!

Quick explanation of fascism: all goals and rhetoric of the State are turn towards itself. What matters under this regime is the successful consolation of every citizen's efforts and allegiance to the Nation. Order is to be attained through obedience and respect of conservative values. Economically, it is similar to a right-wing liberal government (some exceptions regarding the relationship with the State, corporations and unions since the former is regarded as the most vital). Otherwise though, education, civic life, discourse, politics, ethics and most of a person's everyday life demands that the citizen submit to the regime. Often, a cult of personality is instituted with a myriad of nationalistic symbols. The political promise is that this harsh system will transform a moribond, corrupt society into a formidable, orderly Nation under the supervision of a visionary leader. Dissent is impossible. An extreme form of nationalism, if you will.

Nazism is more or less the same, but with one more very important feature: Race. Hitler viewed that the Aryan race was superior to any other. An Aryan is ideally a white skin, blue eye, blond hair, tall and athletic person. He even viewed blacks has not too far behind since the usually more muscular and masculine features. Now, picture fascism with this at its core beliefs and purging the Nation of "impurities" becomes a important item on its agenda. If the Nation is to prevail, it must be full o blondies obediently serving the Man. Communists, handicaps, gays (lesbians were apparently tolerated), political opponents were the usual suspects of the fascist regime. As long as there is someone to blame, the rethoric that the Nation is under attack and corrupted holds. With Nazism, you add the Jewish, the Arabs, the "weak" corrupted whites (read tolerant liberals), the what have you has long as it angers the majority of your supporters. It is the Other's fault if you're not 100% secure, pure and swimming in prosperity (parallels with radicals tea partiers who rant about muslims are scary for that reason: they easily swallow that lie). And, if it wasn't enough, Hitler justified his invasion of Europe to make room for this race to flourish; to establish the "Lebensraum" (Vital/life space) of the Aryans.

TL;DR: Fascism is the absolute obedience to the State to attain perfect Order. Nazism is mostly the same but with a racial component.

Edit: I realize I gave more political explanation than a historic one. I'm on mobile right now, but I can come back with sources later. I suggest that you do ask an historian about the differences in regimes because there was lot of them despite similarities in ideology.

Edit2: Corrected Liebenstraum to Lebensraum.

you might also cross post this to /r/askpolitics