n. 1. In source code, some non-obvious constant whose value is significant to the operation of a program and that is inserted inconspicuously in-line (hardcoded), rather than expanded in by a symbol set by a commented '#define'. Magic numbers in this sense are bad style.
2. A number that encodes critical information used in an algorithm in some opaque way. The classic examples of these are the numbers used in hash or CRC functions, or the coefficients in a linear congruential generator for pseudo-random numbers. This sense actually predates and was ancestral to the more common sense 1.
3. Special data located at the beginning of a binary data file to indicate its type to a utility. Under UNIX, the system and various applications programs (especially the linker) distinguish between types of executable file by looking for a magic number. Once upon a time, these magic numbers were PDP-11 branch instructions that skipped over header data to the start of executable code; the 0407, for example, was octal for 'branch 16 bytes relative'. Nowadays only a wizard knows the spells to create magic numbers. How do you choose a fresh magic number of your own? Simple -- you pick one at random. See? It's magic!