How to understand the notion of boolean query via Immerman's definition A query is any mapping $I:STRUC[\sigma] \to STRUC[\tau]$ that is polynomially bounded. A boolean query is a map $I_b: STRUC[\sigma] \to \{0,1\}$. A boolean query can also be thought as the susbset:
  $$
\{ A \in STRUC[\sigma] \ \mid \ I_b(A) = 1 \}
$$


From this definition of query, used by Immerman in his book "Descriptive Complexity", how are boolean queries and general queries related? It seems from the definition that they are two different things entirely. Is there an intuitive way to link the two concepts? Thank you!
Boolean queries are a special case of general queries. You can take $\tau$ to be some convenient vocabulary and choose a pair of $\tau$-structures to represent "true" and "false".

FOr a  concrete example, take $\tau$ to be the vocabulary with a single nullary relation symbol&nbsp;$T$. Now, associate "true" with the $\tau$-structure $\mathfrak{T}$ that has $|\mathfrak{T}|=\emptyset$ and $T^\mathfrak{T}=\{\langle\,\rangle\}$ (i.e., the nullary relation that contains the empty tuple) and "false" with the structure $\mathfrak{F}$ that has $|\mathfrak{F}|=\emptyset$ and $T^\mathfrak{F}=\emptyset$ (the nullary relation that doesn't contain the empty tuple).

Empty universes are a bit awkward, since they mean that $(\forall x\,\varphi) \rightarrow \exists x\,\varphi$ isn't a tautology. If you don't like that and/or you don't like nullary relations, make the relation&nbsp;$T$ unary and use a one-element universe.