A book that explains what Logic is is P.F. Strawson, Introduction to Logical Theory. The first chapter is devoted to the problem of what Logic and doing Logic is; then there are several chapters presenting various kinds of formal logics and their properties; finally two chapters devoted to the problem of inductive reasoning, what that is and how it essentially differs from probabilistic reasoning (which rather happens to be a mode of deduction).
Strawson, broadly speaking (and for what it's worth), belongs to the Analytic/(post-)Wittgesteinian tradition: the emphasis is on Logic proper (whence the capital 'L'), i.e. the logic of/in natural language, then how and to which extent that has any relation with formal logic.