I am currently learning about Hausdorff measure and dimension. In several of the resources I have encountered, the author provides several different definitions of dimension, and they are often presented as precursors to the Hausdorff dimension. For instance, I have encountered the notion of a dimension of a "self-similar" set:
If a set $E\subseteq X$ can be subdivided into $N\left(s\right)$ subsets that are scaled down versions of $E$ by a factor of $s$, then we define the self-similar dimension of $E$ to be $\dim_{sim}\left(E\right)=\log\left(N\left(s\right)\right)/\log\left(1/s\right)$.
I have also seen the definition of the "box-counting dimension":
For a bounded subset $X$ of $\mathbb{R}^{n}$, partition $\mathbb{R}^{n}$ by a regular rid of cubes of side-length $s$ and count how many of them intersect $X$; if this number is $N\left(s\right)$, then we define the “box-counting dimension” of $X$ to be $\lim_{s\rightarrow0}\log\left(N\left(s\right)\right)/\log\left(1/s\right)$.
The resources I have seen just give the definitions and calculate the dimension of a few sets. What I am struggling to understand is why we are interested in giving these definitions. What exactly are they meant to capture, and what does the "box-cutting" dimension of a set, for instance, tell us about that set?