Definitions
Unit vector has length 1. Orthonormal vectors are orthogonal and unit vectors.
RobJohn's suggestions for the basis in polar coordinates, here, satisfy the criteria but how can you approach this geometrically? And how to make sure they are actually orthogonal in polar coordinates?
$\begin{align} \hat{e}_R&=\frac{(x,y,z)}{r}\\ \hat{e}_\theta&=\frac{(-y,x,0)}{\sqrt{r^2-z^2}}\\ \hat{e}_\phi&=\frac{(-xz,-yz,r^2-z^2)}{r\sqrt{r^2-z^2}} \end{align}$
Perhaps related
Visual Ways to Remember Cross products of Unit vectors? Cross-product in $\mathbb F^3$? (third vector from two other unit vectors)
Explain Dot product with Partial derivatives in Polar-coordinates (orthogonality check)
Example problem (about page 817 here)
$\nabla\cdot\bar{F}=\left(\hat{e}_{R}\partial_{R}+\frac{1}{R}\hat{e}_{\theta}\partial_{\theta}+ \frac{1}{R\sin(\theta)} \hat{e}_{\phi}\partial_{\phi}\right)\cdot\bar{F}$
where
$\bar{F}=R^3 ( \cos(\phi)\sin(\theta)\bar{i}+\sin(\phi)\sin(\theta)\bar{j}+\cos(\theta)\bar{k}).$