A simple method for constructing a nonstandard model of arithmetic is to use compactness as follows. Let $c$ be a new constant symbol not occurring in the language of arithmetic and add to PA the infinitely many new axioms $0<c$, $S0<c$, $SS0<c$, etc. This new theory is consistent, since any finite subset $F$ of it contains only finitely many of the new axioms and has a model consisting of the standard model with the new constant $c$ interpreted as a sufficiently large integer to cover the finitely many new axioms occurring in $F$.
Now take a model of the (whole) new theory -- it is a model of PA and contains an element (the interpretation of $c$ ) which is bigger than all the standard natural numbers $0, S0, SS0$, etc. To be precise, the reduct of this model to the language of PA is the model you want.
Finally, this construction has nothing to do with incompleteness. You could start with complete arithmetic (i.e. all first order sentences true in the standard model) instead of PA and then we'd have proved the existence of non-standard models which satisfy exactly the same sentences of arithmetic as does the standard model.