Understandng euclids theorem Reading this Wikipedia article, it states "If q is not prime, then some prime factor p divides q"
Why does some prime factor divide q? Does mean that for any number there is some prime factor p that divides q?

"Why does some prime factor divide $q$?"
Because if it didn't, then $q$ would itself be prime.
(This of course assumes $q\ne1$.  For Euclid, that was not a problem since Euclid didn't even consider $1$ to be a number.  The question "Why is $1$ not considered prime?" gets answers here that seem to me too complicated.  I'd rather answer it something like this: Suppose we are factoring a number:
$$
\begin{array}{cccccccccccl}
& & & & & & 60 \\
& & & & & \swarrow & & \searrow \\
& & & & 15 & & & & 4 \\
& & & \swarrow & \downarrow & & & & \downarrow & \searrow \\
& & 3 & & 5 & & & & 2 & & 2 & \text{Stop here.} \\
& \swarrow & \downarrow  \\
1 & & 3 & \leftarrow& \leftarrow& \leftarrow& \leftarrow& \leftarrow& \leftarrow& \leftarrow& \leftarrow & \text{Don't do this.}
\end{array}
$$
You can keep factoring out $1$s forever and it gives you no information about the factorization of the number.  Therefore the number $1$ belongs in a different role from those of composite numbers, like $60$, which are the ones to be factored, and prime numbers like $2$, $3$, and $5$, which is where the process of factoring ends.
