I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations":
Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying $P\cap\Omega\neq\emptyset$. A function defined almost everywhere in $\Omega$ is said absolutely continuous on the line $P$ if it is continuous on each closed interval of $P\cap\Omega$.
In particular, this definition was used in the following theorem:
Suppose $u\in L_{\text{loc}}(\Omega)$ and $\frac{\partial u}{\partial x_i} \in L_p(\Omega)$, $p \geq 1$. This function changed on a set of measure zero is absolutely continuous on almost all lines parallelto the axis $x_i$ . Let us denote by $[ \partial u/ \partial x_i ]$ the usual derivative and by $\partial u/\partial x_i$ the distribution derivative. Then we have almost everywhere $[ \partial u/ \partial x_i ]=\partial u/ \partial x_i$ . Conversely, if $u\in L_{\text{loc}}(\Omega)$ is absolutely continuous on almost all lines parallel to the axis $x_i$ with $[ ∂ u/ ∂ x_i ]\in L_p(\Omega)$, then we have $∂ u/ ∂ x_i=[ ∂ u/ ∂ x_i ]$.
There is an explanation in book about the boldface: The set of all intersections of parallel hyperplanes where $u$ is not absolutely continuous, with the hyperplane $x_i = 0$, is a set $M$ such that $\mathcal L^{N−1}(M) = 0$. But what does this really mean? Are there any relationships between this definition and the usual definition of the absolute continuity?
This theorem basically tells you that among "almost every" lines parallel to the axis, the function behaves like an absolutely continuous function.
In another words, this theorem tells you that the Sobolev function behaves very good at the most of points, and the bad points have measure $0$ and hence in some case is ignorable.
Maybe you should compare this result with 1 dimensional case. An one dimensional Sobolev function is absolutely continuous, not bad points; but when dimension rise, the Sobolev function no longer equal to an absolutely continuous function, however, most of it is still can be represented by an absolutely continuous function.
For your second question. This is not a new definition of absolute continuity, but rather a theorem tells you that which part of a Sobolev function can be identified as an absolute continuous function.