The accepted definition of a Prime number is a natural number which has exactly two distinct divisors. The number 1 has only one distinct divisor: itself. The number 0 can be said to have infinitely many distinct divisors.
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The number 1 is neither composite nor prime.
There is no name for numbers which are neither prime nor composite.
Zero and one are neither composite nor prime.
They are neither prime numbers nor composite numbers. It is because, they do not have two factors at least.
A prime number has only 2 factors which are 1 and itself. Composite numbers are everything else except 1 and 0. 1 and 0 are neither prime, nor composite.