What exactly is nonstandard about Nonstandard Analysis? I have only a vague understanding of nonstandard analysis from reading Reuben Hersh & Philip Davis, The Mathematical Experience. As a physics major I do have some education in standard analysis, but wonder what the properties are that the nonstandardness (is that a word?) is composed of. Is it more than defining numbers smaller than any positive real as the tag suggests? Can you give examples? Do you know of a gentle introduction to the nonstandard properties?

To complement the fine answers given earlier, I would like to address directly the question of the title: "What exactly is nonstandard about Nonstandard Analysis?"  
The answer is: "Nothing" (the name "nonstandard analysis" is merely a descriptive title of a field of research, chosen by Robinson). This is why some scholars try to avoid using the term in their publications, preferring to speak of "infinitesimals" or "analysis over the hyperreals", as for example in the following popular books:
Goldblatt, Robert, Lectures on the hyperreals. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998
Vakil, Nader, Real analysis through modern infinitesimals. Encyclopedia of Mathematics and its Applications, 140. Cambridge University Press, Cambridge, 2011.
More specifically, there is nothing "nonstandard" about Robinson's theory in the sense that he is working in a classical framework that a majority of mathematicians work in today, namely the Zermelo-Fraenkel set theory, and relying on classical logic.
