Triode vs Penthod vs Tetrod tube operations.

There are many ways to design a tube amplifier. Some of them use Triode operation, some of them Penthod, some of them Tetrod.

What is the difference between these types of operations in terms of signal flow and/or in terms of circuitry?

Is there any way to get book on some basics of the tube electronics today? Does anyone have the vintage old one?
VTL publishes a book that should answer your questions. Try and contact your local dealer of VTL directly. In the meantime follow this
There are as many books and pamphletes as there are designers. There are a large number of "text-book" style books that were published between the 30s and 60s that would be a good start. Additionally I would recommend that you get your hands on a copy oof the "RCA Receiving Tube Manual", esp. one published after '65~72. In addition to the one by VTL, there is a book "Understanding Tube electronics", by H. Rosenberg, originally published by NY Audio Labs. THere are also a bunch of books about Hi-Fi and Audio published from the Late '50s thru present some of wich go into detail. Also check out Howard Sams publications, I have an "encyclopedia of audio" that has schematics and many chapters devoted to this issue.

The short answer, assuming you are talking about the recent(last 10 years) fad of using an amp in one or the other modes via switching, is a change in class of operation and total power output, w/ Triode being class A, and lowest power, and Pentode being class AB1~2(depending on the circuit, thru D) and highest power.

Also try some of the tech forums on Audio Asylum.
Happy listening