We are able to recognize that they are nonstandard because they contain numbers that we recognize are infinite. But there is absolutely no statement that can be made from within the model from which it could be discovered that those numbers are infinite.
Furthermore, it is possible to construct nonstandard models such that every statement that is true in our model, remains true in that one, and ditto for every statement that is false. They really look identical to our model, except that we know from construction that they aren't. This fact is what makes the transfer principle work in nonstandard analysis, and the ultrapower construction shows how to do it.
(My snark about NSA is that we shouldn't need the axiom of choice to find the derivative of x^2. But I do find it an interesting approach to know about.)
Furthermore, it is possible to construct nonstandard models such that every statement that is true in our model, remains true in that one, and ditto for every statement that is false. They really look identical to our model, except that we know from construction that they aren't. This fact is what makes the transfer principle work in nonstandard analysis, and the ultrapower construction shows how to do it.
(My snark about NSA is that we shouldn't need the axiom of choice to find the derivative of x^2. But I do find it an interesting approach to know about.)