What are access structures in Attribute-Based Encryption?
An access structure is the set of sets that should have access. For example, if the set of parties is $\{P_1, P_2, P_3, P_4\}$ and the access structure, $\mathbb{A}$, is
$$\{\{P_1, P_2\}, \{P_3, P_4\}, \{P_1, P_2, P_3\}, \{P_1,P_2,P_4\},\{P_1, P_3, P_4\}, \{P_2, P_3, P_4\}, \{P_1,P_2,P_3,P_4\}\}$$
then $P_1$ and $P_2$ together have access, $P_3$ and $P_4$ together have access, but $P_2$ and $P_3$ together don't. Note that this access structure is monotonic. Since $\{P_1, P_2\}$ has access, $\{P_1, P_2, P_3\}$, $\{P_1, P_2, P_4\}$, and $\{P_1, P_2, P_3, P_4\}$ all have access.
Yes, it is meant to be the power set. $\mathbb{A}$ is a set of sets of parties.