One is not a prime number because the definition of a prime number is any number that is greater than one that is only divisible by one and itself. Since one is of course not greater than one, it does not fit the definition of a prime number.
A number can either be a prime or a composite number. A composite number has more factors than one and itself. An example of a composite number is 12, which has the factors one, two, three, four, six and 12. An example of a prime number is 11, which has no other factors than one and 11.