Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields). Please specify the exact framework for non-standard analysis you are using in your question (e.g., what definition of "hyperreal number" you are using).
A non-zero element $\varepsilon$ of an ordered field is infinitesimal if $|\varepsilon| < \frac{1}{n}$ for all $n \in \mathbb{N}$. Nonstandard analysis is analysis done in fields with infinitesimals.
There are many ordered fields which contain infinitesimals, but the most common is the hyperreal field. Denoted by ${}^*\mathbb{R}$, the hyperreal field has a subfield isomorphic to $\mathbb{R}$ and is therefore the perfect setting for formalising the arguments of Leibniz and Newton (without the need for limits). This came to fruition in the 1960's thanks to the work of Abraham Robinson.
See also infinitesimals.
