What is the negation of "$A\subseteq B$"?

Unfortunately not. $A \subset B$ means $\forall x \in A, x \in B$, so the negation is $\exists x \in A, x \not \in B$ i.e. there is some element of $A$ that is not in $B$.

Intuititively, since $A \subset B$ means that $A$ is "entirely contained in $B$". $A$ not being a subset of $B$ then means that $A$ is not entirely contained in $B$. This includes the situation that they are disjoint, but also includes the situation where some elements of $A$ are in $B$, but not all of them (i.e. $A$ is partially contained in $B$).

$$A \subseteq B \equiv \forall x( x \in A \rightarrow x \in B)\tag{1}$$

Negating $(1)$ gives us:

$$ \begin{align} A \not\subseteq B & \equiv \lnot \forall x(x\in A \rightarrow x \in B) \\ \\ & \equiv \exists x \lnot(x\in A \rightarrow x \in B) \\ \\ & \equiv \exists x \lnot[\lnot(x \in A) \lor (x \in B)] \tag{$p \to q \equiv \lnot p \lor q$}\\ \\ & \equiv \exists x [\lnot\lnot(x \in A) \land \lnot (x \in B)] \tag{DeMorgan's}\\ \\ & \equiv \exists x [x \in A \land \lnot (x \in B)] \\ \\ & \equiv \exists x (x\in A \land x \notin B) \tag{$A\not \subseteq B$} \end{align}$$

Remember the the definition of subset: if $A \subseteq B$, then every element of A is an element of $B$. The negation of a statements like "every object is " or "all objects are" is the statement "at least one object is not." Thus, $A \not \subseteq B$ means that at least one element of A is not an element of $B$.