How to understand pushdown automata intuitively?

What is an intuitive way of understanding what a push down automaton is capable of computing?

Intuitively, a pushdown automaton uses a stack and using a stack we can do a depth first traversal of a parse tree. It means that we can accept strings which are in context free languages by using stack of a pushdown automaton (left-most derivation). This is not a rigorous proof that languages of pushdown automata are context free. For proofs you must see various textbooks on the subject.

StackExchange Computer Science Q#56462, answer score: 4