n-skeleton

Remarks

The torus $T^2$ has a CW structure with one $0$-cell, two $1$-cells $a$ and $b$, and one $2$-cell attached along $a b a^{-1} b^{-1}$. Its $0$-skeleton is a point, its $1$-skeleton is a wedge of two circles, and its $2$-skeleton is $T^2$.

The skeleton depends on the cell structure and not only on the space. The sphere $S^2$ with one $0$-cell and one $2$-cell has a point as $1$-skeleton. The sphere $S^2$ as the boundary of a tetrahedron has as $1$-skeleton a graph with $4$ vertices and $6$ edges. So "the $1$-skeleton of $S^2$" has no meaning before a cell structure is chosen.

For a CW complex, $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells along maps from their boundary spheres, and $X$ is the union of its skeleta with the weak topology. The inclusion $X^{(n)} \to X$ induces an isomorphism on homotopy groups $\pi_k$ for $k < n$ and a surjection for $k = n$.