Cantor's Diagonal Argument is a mathematical proof that demonstrates the set of real numbers is uncountably infinite, meaning it cannot be listed in a complete sequence. The argument shows that for any attempt to list all real numbers between 0 and 1, one can always create a new real number not included in the list by changing the digits along the diagonal of the list. This method highlights that there are different sizes of infinity. While the set of natural numbers 1, 2, 3, ... is countably infinite, the set of real numbers is larger, illustrating that some infinities are more significant than others, a concept introduced by Georg Cantor.