Let number mean positive integer.
What makes a number perfect is quite intricate:
A perfect number is a number that is equal to the sum of its proper divisors.
What makes a number a power is also quite intricate:
A number $N$ is a power (number) if there exist numbers $n, k$ such that $N=n^k$.
But what makes a number a perfect power is quite boring:
A power number $N=n^k$ is perfect if $n,k > 1$.
I wonder why so much ado is made when calling a (non-trivial) power "a perfect power" ("Ah, $16$ is a perfect power!"), and not just "a power".