Is this language countable : $L= \{ w : w \in (1 + 0)^{*} \}$
The language $$L= (0+1)^*$$ (the set of all strings over $0$ and $1$) is countable. Furthermore, any subset of $L$ is also countable. However, the set of all sublanguages of $L$
$$S = P(L) = \{M \mid M \subseteq L\}$$ (a set of sets in fact) is not countable. This video proves this using the diagonalization argument. 

$S$ and $L$ are two different sets. In fact, $L$ is a member of $S$.