The Luhn algorithm was created by Hans Peter Luhn and is a way of creating a simple checksum for a number. This algorithm, also known as the mod 10 algorithm, is used in a wide variety of applications but is commonly associated with credit card numbers.
If you look at the numbers on the front of your credit card the last digit on the right is the checksum. An algorithm is done on the other numbers and if the checksum is the same then the number is considered valid.
Outside of credit card numbers, the Luhn algorithm can be used to create a checksum on any number that you want to store. It is especially handy when you want to give users a number that they will be hand typing into a computer. The checksum helps spot any errors in typing in the number before that number is processed. The good thing about the Luhn algorithm is that it doesn't matter how long the number is so it will work with any kind of digit sequence.