Non-standard models of arithmetic are structures that satisfy the axioms of Peano arithmetic (PA) but contain "non-standard" elements that do not correspond to the standard natural numbers (0, 1, 2, ...). In other words, while a standard model of arithmetic consists only of the usual natural numbers, a non-standard model includes additional "infinitely large" and "infinitesimally small" numbers that do not have a counterpart in the standard model.
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A nonstandard model of Peano arithmetic is a model not isomorphic to the standard natural-number structure. It contains the standard numerals as an initial segment and also contains elements larger than every standard numeral.