In my book Sobolev's spaces are defined as follows
$$W^{m,p}(\Omega) = \{v \in L^p(\Omega): \,\, D^{\alpha}v \in L^p(\Omega) \,\, \forall \, |\alpha|\leq m \,\mbox{ in the sense of distributions} \}$$
I don't understand what he means by distributional sense. To be more specific in an example, what does mean $u \in L^2 (\mathbb{R}) $ in the distributional sense?
The phrase "in the sense of distributions" here refers to the differential operator $D^\alpha$. For example, $u \in W^{1,1}(\mathbb{R})$ if $u, Du \in L^1 (\mathbb{R})$, where $Du$ is the weak derivative, or derivative in the sense of distributions. This derivative is computed by "integrating" against an arbitrary smooth function with compact support $\phi$.
\begin{aligned} \langle Du, \phi \rangle &= \int u^{'} \phi \; dx \\ &= -\int u \phi ^{'} \; dx \end{aligned} It's a way of extending the notion of differentiability to functions that are not necessarily continuous.