Does a term being normalizable mean the same as the term has a normal form?
Yes, a term is normalizable if, and only if, it has a normal form.

It is well known that $\omega = (\lambda x . x x) (\lambda x . x x)$ is not typeable, nor is it normalizable. It is not normalizable because it has an evaluation step ($\beta$-reduction) which evaluates it to itself, so we get $\omega \mapsto \omega \mapsto \omega \cdots$.

To see that it is not typeable, observe that already its subexpression $\lambda x . x x$ is not typeable. For if $x$ has type $T$ then in order for $x x$ to have a type, $T$ must be of the form $T \equiv T \to S$ (because $x$ is applied to something, therefore by the typing rules it must have a function type). But there is no type $T$ such that $T \equiv T \to S$, because the type on the right-hand side is larger (as a syntactic expression) than the type on the left-hand side.