Why are all non-prime numbers divisible by a prime number? In Euclid's infinite prime numbers proof, the logic is as follows:
Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes)
Then there must be a greatest prime $p$
$$n = (2 \cdot 3 \cdot 5\cdots p) + 1$$
$n > p$, and under the proof's assumption, $n$ cannot be prime.*
This is where the logic confuses me. Why is it that given that the if a number is not prime, then it is automatically divisible by a prime. I can't think of an example to contradict, but that's not proof that there exists no number that is not prime and non divisible by primes.  

You are in good company in your error. Some great mathematicians, including Dirichlet, have made the same mistake: falsely reporting that Euclid's proof was by contradiction.
Euclid's proof says that if you take any finite set of prime numbers (for example, $2$, $11$, and $19$) and multiply them and then add $1$, the resulting number is not divisible by any of the primes in the finite set you started with (thus $(2\cdot11\cdot19)+1$ is not divisible by $2$, $11$, or $19$ because its remainder on division by any of those numbers is $1$.
Therefore, the finite set you started with can be extended to a larger finite set: the prime factors of (in this example) $(2\cdot11\cdot19)+1$.)
The reason the number $(2\cdot11\cdot19)+1$ must be divisible by some prime is that if it is not divisible by any prime other than itself, then it is prime and it is of course divisible by itself.
PS: Some people commenting below are unhappy with my last paragraph above, so I'll add this: Let's do a proof by contradiction on this one: Consider the smallest number $N$ that is not divisible by any prime.  It cannot be divisble by anything smaller than itself except $1$, since that not-necessarily-prime factor, being smaller than the smallest counterexample, would be divisible by some prime, and then $N$ would be divisible by that prime.  So not being divisible by anything except itself and $1$, $N$ would be prime, and hence divisible by some prime, namely itself.
