TITLE: What is an example of a non standard model of Peano Arithmetic?
QUESTION [25 upvotes]: According to here, there is the "standard" model of Peano Arithmetic. This is defined as $0,1,2,...$ in the usual sense. What would be an example of a nonstandard model of Peano Arithmetic? What would a nonstandard amount of time be like?

REPLY [7 votes]: This is a historical footnote to the nice accepted answer above. More precisely, the method in the yellow box in 

seems to be already hinted at in Kurt Gödel's review in Zentralblatt, Band 7, Heft 5, of 

Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems. 
  Norsk Mat. Forenings Skr., II. Ser. No.1/12, 73-82 (1933).

Strictly speaking, Gödel does not give an argument in the review, only says that Skolem's result follows easily from his own article on the incompleteness theorem; here is Gödel's review of Skolem's paper, with Gödel's hint highlighted in orange (the right-hand side is my translation):

(the above picture larger)