What exactly is a logic?
Fundamentally, a logic consists of two things.


Syntax is a set of rules that determine what is and is not a formula.
Semantics is a set of rules that determine what formulae are "true" and what are "false". To a model theorist, this is expressed by relating formulas to the mathematical structures that they're true in; to a proof theorist, truth corresponds to provability from a chosen set of axioms with a chosen set of proof rules (techniques).


The difference between different logics is, most simply, in the choice of syntax and semantics. Most logics are extensions of propositional logic or first-order logic. In a sense, you can see these extensions as "adding more features" to the logic. For example, temporal logics deal with truths that can vary over time.

In general, these features could be expressed in a simpler logic, at the cost of having to write longer formulas. For example, the temporal concept "$\varphi$&nbsp;is true from this point for eternity" can be expressed in a first-order way by adding a time parameter to all your propositions and saying "For all times&nbsp;$t$, if $t$&nbsp;is greater than or equal to the current time, then $\varphi$&nbsp;is true at time&nbsp;$t$."  In a sense, you can think of these logics as adding libraries to a basic programming language so you can say things more easily.

Since pretty much all logics are based on propositional and first-order logic, I'd recommend learning about those first.