Designing a Pushdown Automation to accept a language

Im a novice trying to understand the theory of computation.Im trtying to learn about PDA.I understand that it is a machine counterpart of CFG.Im basically referring to Introduction to Automata Theory by John E. Hopcroft and Jeffrey D.Ullman and this video http://youtu.be/8O6FUFwWt5A?t=6m32s by IIT Professor Kamala

I understand the plates placed above spring analogy.What confuses me is that the reference example.It uses a table like the one shown here http://youtu.be/8O6FUFwWt5A?t=6m32s

Where do we get this table and can i use this table for all languages?

Like if a question comes up like this

Can i use the same table and what im supposed to do? I know it might sound stupid,but please help me,im trying to learn.

• In this example, you can use the stack to store the expected number of 'b's, but when you are done with this, try $\{a^{2n}b^n: n\geq 1\}$. Then you will be able to solve $\{a^{pn}b^{qn}: n\geq 1\}$ for some fixed positive integers $p,q$. – user1537366 Dec 8 '14 at 13:23

Try to analice the language you want to build a PDA for. Have you seen the standard example for the language $\{a^nb^n\mid n\in\mathbb N\}$? (I guess this should be in the book you cite.)
What would you do to implement a PDA for this simpler example? What are the states and what is the role of the stack in this example (it is used to store $a$'s so you can count/determine whether the number of $b$'s matches with the number of $a$'s).
Now for the example you are interested in, what is the difference (now the number of $a$'s and $b$'s is not the same, but there is still a strong connection between them). Can you modify the PDA for the language $\{a^nb^n\mid n\in\mathbb N\}$ to build one for your example?