For two Riemann integrable, $2\pi$-periodic functions $u$ and $v$, we define
$$(u,v)=\frac{1}{2\pi}\int_{0}^{2\pi} u(x)\overline{v(x)} dx$$
where the complex number $(u,v)$ is called the scalar product of the functions $u$ and $v$.
Is is not at all clear to me why this definition works? What is the reason for multiplying one function/vector by the conjugate of another function/vector? And then why would we want to integrate the result of that multiplication?
On
Multiplying by $\overline{v(x)}$ instead of just $v(x)$ ensures the positivity property of inner products:
$$ (u, u) = \frac{1}{2\pi} \int_{0}^{2\pi} u(x)\overline{u(x)}\:dx = \frac{1}{2\pi} \int_{0}^{2\pi} |u(x)|^2\:dx \ge 0. $$
Inner products must output scalar values, so integrating is a way getting this while maintaining the properties (positive-definiteness, additivity and homogeneity in the first slot and conjugate symmetry) and extracting useful information of the functions behavior on a certain interval. We could define an inner product which would be easier to calculate, such as
$$ (u, v) = u(a)\overline{v(a)} $$
for some fixed constant $a$. But this would only give us information about how $u$ and $v$ behave at the fixed point $a$ instead of a whole interval and wouldn't satisfy the definiteness property because we could have $u(a) = 0$ and then $(u, u) = 0$ with $u \neq 0$.
The definition works because it fulfills the criteria in the definition of scalar product / inner product. It also works because it's a very natural extension of the standard inner product on finite complex vector spaces.
We conjugate because we want the scalar product of a function with itself to be real (and positive, as long as the function isn't $0$). We integrate because that's one of the most natural ways to assign a single number to a function.