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What does it mean for characters to be orthogonal?

For example, if we take the following definition of the Orthogonality relation for characters: $$ \sum_{r = 1}^{n}\overline{f}_r(a_i)f_r(a_j) = \begin{cases} n, & \text{if $a_i$ = $a_j$}\\ 0, & \text{if $a_i \neq a_j$} \end{cases} $$

As defined in 6.12 of Apostol's Introduction to Analytic Number Theory.

My question is about the syntax $\overline{f}_r(a_i)$. What does character orthogonality actually look like? Or is the definition literally just any function $\overline{f}$ such that it fulfils the above relationship?

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    See the [orthogonality relations](https://en.wikipedia.org/wiki/Character_theory#Orthogonality_relations) – 2017-02-21

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Here $\bar{f}_r(a_i)$ simply denotes the complex conjugate of $f_r(a_i)$. So there is no separate function $\bar{f}$ in this equation; it's just an equation about the functions $f_r$.