Reading some potential theory, my book almost always uses a rather strong regularity condition on the boundaries to be of class $C^2$. My book refers to a slightly weaker case of Lyapunov boundaries. A boundary is Lyapunov if at each point $x\in \partial D$ the normal vector $\nu$ exists and there are positive constants $L$ and $\alpha$ such that the angle $v(x,y)$ between satisfies $v(x,y) \leq L\|x-y\|^{\alpha}$ $\forall x,y\in \partial D$.
Only intuition behind this definition I have is that the angle has to approach $0$ faster than $L\|x-y\|^{\alpha}$ as $x\rightarrow y$. I am having trouble coming up with smooth boundaries which would not be Lyapunov. What are some canonical uses of Lyapunov boundaries in literature?
If the boundary can be locally parametrized by a $C^2$-smooth diffeomorphism, then the condition holds. More generally, it holds if there is a $C^1{,\alpha}$ parametrization. (Such domains are usually called simply "$C^{1,\alpha}$ domains".)
A simple example is the planar domain $$D= \{(x,y): |x|^{1+\epsilon}+|y|^{1+\epsilon}<1\}$$ Its boundary is $C^{1,\epsilon}$ smooth, but not $C^{1,\alpha}$ smooth for $\alpha>\epsilon$. The normal vectors near $(1,0)$ are such that $v(x,y)$ is of the order of $\|x-y\|^\epsilon$.
As an amplification of the above example, take $$D= \{(x,y): |x|/(1+|\log x|)+ |y|/(1+|\log y|)<1/10 \}$$ This is a $C^1$ domain which is not $C^{1,\alpha}$ for any $\alpha>0$. The condition on $v(x,y)$ fails for every $\alpha>0$.
As for where this condition is used: mostly in the potential theory and in the regularity theory of PDE, especially elliptic ones. The solution of a boundary value problem may sometimes be represented by the integral over the boundary (involving the boundary kernel), and the rate at which the normal changes may affect the degree of smoothness of the solution on the boundary. The book on Elliptic PDE of 2nd order by Gilbarg and Trudinger has both positive results and counterexamples on this topic.