In the definition of a CW complex we construct a topological space by gluing $n$-cells to the $n-1$-skeleton, and then we give to it another topology (it already has one: the one given by the gluing operation which is a quotient topology) choosing a particular family of closed sets as fundamental covering (or equivalently the initial topology given by the characteristic maps).
I imagine that the two topologies (the quotient and the weak one) are different (otherwise why call it "weak"? It would be enough to define the CW complex as a gluing of cells), but I would make good use of an example. Can anyone help me? Thanks
The two topologies are exactly the same, as you can easily prove by induction on $n$. Indeed, supposing the two topologies are the same on the $(n-1)$-skeleton $X^{n-1}$, the quotient topology on $X^n$ has the property that a map out of it is continuous iff its restriction to $X^{n-1}$ is continuous and its composition with the characteristic map of each $n$-cell is continuous. But by induction, the restriction to $X^{n-1}$ is continuous iff the composition with the characteristic map of each cell of dimension $<n$ is continuous. This shows the quotient topology has the same universal property as the weak topology.
So there is no need to refer to the weak topology induced by the cells in defining a CW-complex, and some authors do not do so (for instance, Hatcher's Algebraic Topology just defines the topology as the quotient topology). For infinite-dimensional CW-complexes, you can define the topology as the colimit topology with respect to the inclusions of the skeleta, where the skeleta are inductively given the quotient topology.
As for why the topology is called "weak", "weak topology" is sometimes used as a synonym for "colimit topology" (but beware in functional analysis it is instead used as a synonym for "limit topology"!). In particular, for instance, the weak topology on an infinite CW-complex is typically much finer than other natural topologies that may exist. For instance, a wedge of countably infinitely many circles can be given a topology by identifying it with the Hawaiian earring, a subset of $\mathbb{R}^2$, but the CW-complex topology is much finer than the Hawaiian earring topology.