The ancient Greeks regarded 1 as a prime.
Today it is not.
Mainly because a lot of important theorems would be false
if 1 were a prime.
I'll buy that.
The more modern way of classifying integers is
0 additive identity
+/- 1 units
primes
composites
When you get into algebraic number theory, you deal with things like the so-called Gausian integers, which are complex numbers whose real and imaginary parts are both integers.
Then the classification is
0
+/-1, +/-i
primes
composites
The real reason 1 isn't counted as prime is that would make the unique factorization theorem more complicated:
every integer is the product of a unit and a bunch of primes, and in one way only (except for the order of the factors)
as opposed to
every integer is the product of a bunch of primes, and in one way only (except for the order of the factors, and the fact that any number of 1's can be multiplied in, and any even number of -1's)