Book about different kinds of logic. I'm searching for a book that talks about different kinds of logic (esoteric and particular ones too) and their uses and differences.
Does such a book exist?
 A: There are plenty non-esoteric, but "less classical" logics.
For example, logics in which you can have infinitely many conjunctions, or infinitely many quantifiers, there are other logics in which you add quantifiers (which may or may not be definable in second-order logic, but usually not definable in first-order logic), these may include things like "For all but finitely many ..." or things like $Qxy\varphi(x,y)$ indicates that $\varphi$ defines a linear order, and so on and so forth.
Many of these are covered in the book Model Theoretic Logics (eds. Barwise, Feferman) which is available freely.

Another good point to start with would be Stanford Encyclopedia of Philosophy: classical logic. Yes, this is about classical logic, but in the related entries section you can find many non-classical which are more or less esoteric, including references and possible internet resources.
A: If by esoteric and particular you mean that the book is based solely on a single non-classical logic, then here are some examples:
Fuzzy Equational Logic (Belohlavek, Vychodil), 
Modal Logic (Blackburn)
If you want something that simply exposits many different non-classical logics and says a little bit more about non-classical logics in general, there's "An Introduction to Non-classical Logic - From If to Is" by Priest.
A: There's also Deviant Logic, Fuzzy Logic by Susaan Haack. It is an expanded/updated edition of her earlier, very short and readable (=non-technical) Deviant Logic, which despite the title is not I think hostile to the idea of alternative logics. But I gather she's more sympathetic to older, multi-valued approaches than to Zadeh-style fuzzy logic. EDIT: I take no position on the merits of Haack's views or arguments. But the book is useful as a first dip in the pool, and there's an appendix with a handy summary of the properties of some historically notable systems.
A: You might be interested in Eric Schechter's 2005 book Classical and Nonclassical Logics. Besides, I'm mentioned in the book, so it can't be all that bad!
A: Here's an interesting book which covers different kinds of logical systems (2-valued, 3-valued, infinite-valued).
Also, Zeman's book on modal logic covers a few different modal logics.
A: Focusing on the propositional situation, Humberstone has a book The Connectives of which I'm quite fond. For what it is it is extremely readable; additionally, it can double as a coffee table if you need one.
