Why is weak topology considered weak while others are strong. I know for a banach space $\mathbf{X}$ that the norm $||\cdot||$ produces a topology for the space. I also know this is considered "strong". I also understand that the dual, $\mathbf{X}^\ast$, is the set of all continuous linear functionals. I also understand that the set of functions in $\mathbf{X}^\ast$ can be used to create a topology on $\mathbf{X}$ such that they are all continuous. This one being called "Weak topology", I don't understand this, why is it called weak? Why is it viewed as being "weaker" than the norm based one? What is the motivation behind this?

We speak of weak/strong topologies and also of coarse/fine topologies. Either metaphor relates to the same relationship.
A topology on a set $X$ is a subset $\tau$ of $\mathfrak{P}X$ - the power set of $X$. To qualify as a topology the collection $\tau$ must satisfy the usual axioms.
If $\sigma$ is another topology on $X$ then to say $\sigma$ is weaker than $\tau$ means exactly that, as sets
$$
\sigma \subseteq \tau.
$$
You may find it a useful exercise to explore how this relation induces a partially-ordered set structure on the topologies on $X$, and whether you can find suitable definitions of $\sigma \land \tau$ and  $\sigma \lor \tau$ which make this poset into a lattice.
