The definition of socialism has as many variances and sub plots as does liberalism, conservatism, libertarianism, Marxism, or Nazism.
Technically:
1. A political theory advocating state ownership of industry.[Wordnet]
2. An economic system based on state ownership of capital.[Wordnet]
3. A theory or system of social reform which contemplates a complete reconstruction of society, with a more just and equitable distribution of property and labor. In popular usage, the term is often employed to indicate any lawless, revolutionary social scheme.[Websters].
Dictionary - Definition of socialism
So if you go by the strict definition, Nazi Germany was not socialist in that it did allow for private ownership of property and means of production. It WAS socialist, however as the government controlled the means of production, dictated who would be allowed to own private property, and dictated what wages and benefits the people would receive.
Whenever the U.S. government presumes to redistribute wealth, that is a socialist concept. When the government assumes the power to take the property of one person and give it to another, that is also a socialist concept and starts moving into the realm of totalitarianism.
And yet, despite exercising some of the same concepts here and there, the U.S. government is not anything like the Third Reich. At least not yet.