An Armstrong number is a special type of number in mathematics where the sum of its digits, each raised to the power of the number of digits, equals the number itself. These numbers are often used in programming exercises to teach loops, conditionals, and number manipulation. They are also known as narcissistic numbers and exist in various digit lengths, such as 3-digit (153, 370) and 4-digit (9474) examples. Armstrong numbers are rare and interesting due to their unique mathematical properties.