Loog at ASCII wikipedia.
The American Standard Code for Information Interchange (ASCII /ˈæski/ ass-kee)1 is a character-encoding scheme originally based on the English alphabet that encodes 128 specified characters - the numbers 0-9, the letters a-z and A-Z, some basic punctuation symbols, some control codes that originated with Teletype machines, and a blank space - into the 7-bit binary integers.
So In the ASCII Scheme the '0' char is the number 48, the char '1' is 41, and so on. So c - '0' is equivalent to c - 48. If c is '1' the expression became 49 - 48 = 1. So in few word 'c' - '0' convert a digit char ['0'-'9'] into an integer [0-9].
Edit 1
As suggested by @Eric Postpischil, ASCII is not part of ANSi C (nor c++). But is very common and all compiler I know use ASCII set.
