Answer
$a$ has to be a negative number, and $n$ has to be an odd number.
Work Step by Step
If $a$ is positive, $a^{n}$ will always be positive.
For $a^{n}$ to be negative, $a$ has to be a negative number.
Furthermore, a negative number raised to an even power is always positive.
For $a^{n}$ to be negative, $n$ has to be an odd number.
Both conditions must be satisfied.