2. 24, 224.

The total number of one-to-one mappings from subset $A$ to $B$ is the number of permutations of 3 elements taken from $B$, which is $P_{4}^{3}=24$.
If $A$ is an $m$-element set and $B^{*}$ is an $n$-element set, then the total number of all possible mappings from $A$ to $B^{*}$ is $n^{m}$. Since the mapping from $A$ to the subset of $B$ cannot be an empty set, the total number of mappings is $1^{3} C_{4}^{1}+2^{3} C_{4}^{2}+3^{3} C_{4}^{3}+4^{3} C_{4}^{4}=224$.
2. Let $A=\left\{a_{1}, a_{2}, a_{3}\right\}, B=\left\{b_{1}, b_{2}, b_{3}, b_{4}\right\}$, then the total number of one-to-one mappings from $A$ to the subsets of $B$ is $\qquad$ and the total number of mappings from $A$ to the subsets of $B$ is $\qquad$.