This question may have more of a vague, less objective answer than usual for this site, so I apologise if it difficult to answer definitively.
Below, $p$, $q$ and $r$ are distinct primes.
A group of order $p$ is simple (these groups are precisely the prime-order cyclic groups.)
A group of order $p^n$ is not simple, for $n \geq 2$. (Here is a proof from this site.)
A group of order $pq$ is not simple. (Proof.)
A group of order $p^2q$ is not simple. (Proof.)
A group of order $p^2q^2$ is not simple. (Proof.)
A group of order $pqr$ is not simple. (Proof.)
It is not always possible to classify the simplicity this way - consider $p^2qr$. $60=2^2 \cdot 3 \cdot 5$. $A_5$ is simple, $\mathbb{Z}_{60}$ is not.
Is there a more general statement that can classify whether a group is either a) definitely simple, b) definitely not simple, c) could be either simple or not simple, in terms of the decomposition of the prime factors of the order, or certain results that rule out many more cases?
Probably the most important result here is the Feit-Thompson Theorem (or Odd Order Theorem):
Theorem (Feit & Thompson, 1963). Every finite group of odd order is solvable.
Corollary. Any non-abelian finite simple group has even order.
That is, if $G$ is simple but non-cyclic, then $2$ is necessarily a prime factor of $|G|$.
This result played a significant role in the Classification of finite simple groups. Its proof is notoriously long.