Volume 2 contains chapters 11 to 20 and plenty of bonus pages and translation notes.
Again, this book is a 16+. You don't want your children to read this and learn how cruel life is and become suicidal... or something.
Sayonara, Zetsubou-Sensei revolves around a teacher called Zetsubou-Sensei (real name Nozomu Itoshiki) and his class of unique students. Zetsubou-Sensei is one of the most depressed people you will ever read about. All he wants to do is die. His strange personality is somehow attractive to the females in his class such as a super positive girl, a hikikomori, a stalker, a foreigner, a quite girl, a perfectionist, a girl with a tail fetish, an illegal immigrant and more! Every single character (apart from the ordinary girl who says she isn't ordinary) is really strange!
Basically, this manga is about a class full of strange people who talk about everyday things but delve deeper into the truth and make them seem really odd/bad/funny.
Anyway, with most of the introductions out of the way (see volume 1) we start learning a bit more about why sensei wants to die so much. From the truth of the universe to report cards and unstable weather, just about anything can make someone sink into despair.
We also meet some more new characters in this volume. The chairman who lacks presence, a manga loving girl who's addicted to coupling and sensei's family.
To be able to fully understand Sayonara Zetsubou-Sensei volume 2, you need a little bit of japanese culture knowledge although the translation note pages cover most of the references.