Not an academic, but I have a respectable source on hand and some knowledge of the anarchist movement.

Goldman did work frequently with individual socialists, and the American Socialist Party was relatively powerful during the time, but she was not a socialist herself. She disavowed the party and accused them of being too afraid of 'spook' accusations and following the "crooked path" of politics. In Socialism: Caught in a Political Trap, she said "if once economic power were added to the already supreme political power of the state, its iron heel would cut deeper into the flesh of labor than that of capitalism today."

Goldman was communal (but distinctly not a Marxist Communist), and strictly anarchistic, but refused to clearly state "an iron-clad programme or method on the future," seeing anarchism as naturally developing past theoretical construction. She was generally more an activist than a thinker or writer, and perhaps your teacher took her association with individual socialists as proof of her beliefs.

EDIT: I do want to say something about the language. Anarchism, socialism and communism are all deeply interrelated historically, and there and many blurry areas of description. Classification-wise, libertarian socialism is often used to describe the (very broad) anarchy-as-economic-liberation model which anarcho-communism, anarcho-syndicalism, etc. could all be justly classified as, so it would not be inaccurate to describe Goldman as a libertarian socialist. It would be inaccurate to call her a socialist in an unqualified sense, however, as this usually refers to state-owned means of production as advocated by the democratic American Socialist Party and other contemporary socialist parties.

Source: Demanding the Impossible: A History of Anarchism, Peter Marshall, Ch. 24, Ch. 2.

The term anarchism is really easily misunderstood, partly because there are individualist anarchists and collectivist anarchists who really seem to have little in common, while collectivist anarchists and socialists seem to overlap a lot. As the previous poster wrote, these movements were interrelated and developed over time, with a lot of branching and splitting of participants.

If you focus on the core idea of anarchism being a resistance to and suspicion of government authority, it might be easier to see how a collectivist anarchist differs from socialists: they both want greater equality, they both see capitalism as exploitative, etc. Socialists are either more willing to envision a government-organized socialist future and/or interested in making that future come sooner through, say, revolution..even something like a "dictatorship of the proletariat," which was Lenin's path. As socialists in western Europe and the US started to see what was happening to socialism in the Soviet Union, some concluded that it was very, very dangerous to try to enact socialism through government intervention. They lean more anarchistic. Socialists who thought the main problem with Soviet-style "socialism" was not that it was government-led, but that the government in question was a dictatorship instead of democratic, became associated with "democratic socialism." I'm generalizing a bit, but those are the chief dividing lines.

There's a new book on Emma Goldman and (another anarchist) Alexander Berkman by Paul Avrich, the greatest historian of anarchism. He died before finishing the book, but it was recently completed by his daughter. I haven't read it yet, but all accounts so far are saying it's very good. It's called Sasha and Emma.