The square of a prime is one greater than a multiple of 24? I read that the square of any prime number, excluding 2 and 3, is one greater than a multiple of 24. Is this a conjecture or a theorem? It's hard for me to imagine how such a thing could be proven.

You want to show that $$p^2-1 =24k$$ for some integer k.
$$ p^2-1 = (p-1)(p+1)$$
since $p$ is a prime different from $2$ and $3$, $p$ is odd therefore both $p-1$ and $p+1$ are even and one of them is a multiple of $4$.
That is the product $(p-1)(p+1)$ is a multiple of $8$. 
On the other hand $P=3k+1$ or $p=3k-1$ and in either case the product $(p-1)(p+1)$ is a multiple of $3$.
That makes the product a multiple of $24.$    
