What are p-adic numbers?
The rational numbers (all the positive and negative ratios of counting numbers) form a dense continuum on the number line, but not a "closed set". Effectively there are "holes", like the well-known irrational numbers π, e, √2 and φ (the golden mean). There are in fact infinitely many such "holes", and the most familiar way to "fill them in" is to "complete" the rational numbers in order to arrive at the continuum real numbers, which include all rational and irrational numbers, a closed set.
However, in the 1890s it was discovered that there are infinitely many other ways to "complete" the set of rational numbers, and each of these produces a radically different number system. There turns out to be one for each prime number, and they've become known as the 2-adic, 3-adic, 5-adic, 7-adic, etc. number systems. They key is to change the way you define the distance between two rational numbers. The familiar way is to subtract the smaller number from the larger, to get the kind of distance associated with conventional measurement. The "p-adic" approach (where p stands for an arbitrary prime) involves looking at that same difference (a rational number, assumed to be in reduced fraction form) and then the power of p that appears in its numerator or denominator. In this way, two rational numbers which might be considered "nearby" in the real number system can be vast distances apart in a p-adic system (and vice versa).
A more detailed explanation of p-adic numbers can be found here.