Pushdown automaton accepting $\{a,b\}^*$ with twice as many $a$s as $b$s
When in doubt, one should always consult the definitions.

In this case, this is Definition 7.1 of the book by Peter Linz.

It states that $$\delta : Q \times (\Sigma \cup \{ \lambda \}) \times \Gamma \rightarrow \textrm{finite subsets of } Q \times \Gamma^*$$
Thus, $\delta(q,a,A) = \{[q_1,w_1], \ldots, [q_k,w_k\}$ means that when a pushdown automaton is in state $q$ and sees an $a$ on input (where $a$ may be an input or the empty string $\lambda$), then it can enter any one of the states $q_1$ to $q_k$ and replace $A$ by the corresponding string of stack symbols $w_i$.

Therefore, a pushdown automaton can push a string $w$ to the stack by reading $\lambda$ and replacing it by $w$. Or it can pop the top symbol $b$ by seeing $b$ on the stack and replacing it by $\lambda$.