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A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime are related to proximity in the so called "p-adic metric." . . . . . .
Wikipedia: p-adic number
Mathworld: p-adic Number
Fact archive: P-adic
A. J. Baker 2007/08/12: An Introduction to p-adic Numbers and p-adic Analysis (pdf, mirror) rigorously develops a body of theorems.
David A. Madore 2000/12/07: A first introduction to p-adic numbers (pdf, mirror) is a very well-done and simple introduction to the topic.
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The webmaster and author of this Math Help site is Graeme McRae.