A prime number has to have the property that not every number is a multiple of it. In the language of abstract algebra this means that the ideal generated by that number is proper (not the entire set of numbers one is considering).
Another reason for discounting 1 as a prime number is that considering 1 to be a prime destroys the uniqueness of prime factorization. For instance, 24 is uniquely, up to order of powers of prime factors:
2^3 times 3
If we allow 1 to be prime then I can write 24 as
1^17 times 2^3 times 3
or I can write it as
1^2 times 2^3 times 3
We lose the property that representations of numbers as products of powers of primes is unique. Thus we have a good reason to discount 1 as being prime and there aren't any good reasons to count it as a prime.
The definition given in elementary school is not the correct one. It's a working definition that works well and is correct for all positive integers except 1. In my experience people remember the definition of: "only divisible by itself and 1". Then they ask, "Why isn't 1 considered prime then?" It's because the definition given isn't correct.