Generally, by logic, people mean first order logic, a formal set of rules for building mathematical statements out of symbols like $\neg$ (negation) and $\rightarrow$ (implication) along with quantifiers like $\forall$ (for every) and $\exists$ (there exists).

More generally, a logic is any set of rules for forming sentences (the logic's syntax) together with rules for assigning truth values to them (the logic's semantics). Normally it includes a (possibly empty) set of types $T$ (also called sorts), which represent the different kinds of objects that the theory discusses (typical examples might be sets, numbers, or sets of numbers). In addition it specifies particular quantifiers, connectives, and variables. Particular theories in the logic can then add relations and functions to fully specify a logical language.