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 (hard-coded), 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 pseudorandom 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; 0407, for example, was octal for "branch 16 bytes relative". Nowadays only a wizard knows the spells to create magic numbers. MS DOS executables begin with the magic string "MZ".
*The* magic number, on the other hand, is 7+/-2. The paper cited below established the number of distinct items (such as numeric digits) that humans can hold in short-term memory. Among other things, this strongly influenced the interface design of the phone system.