First-order logic is the rules that determine which propositions follow from other propositions.
More rigorously we can say that "(classical) first-order logic" consists of (1) a set of rules for what well-formed formulas are, given a particular non-logical vocabulary, and (2) a transitive(ish) relation $\vdash$ between finite sets of wffs and wffs, intuitively denoting "this formula can be derived from these other formulas".
There are several different ways of defining the $\vdash$ relation of classical first-order logic, such as by a Hilbert system, or by natural deduction or sequent calculus (or for that matter we can define it to be identical to semantic entailment after developing some rudimentary model theory). These all define one and the same logic, at least according to one way of looking at it.
A first order theory is a particular set of axioms whose consequences (according to first-order logic) you're interested in.
In other words, a "first-order theory" consists of (a) a first-order vocabulary, from which the generic first-order logic rules derive a concept of "well-formed formula", and (b) a particular set of well-formed formulas over that vocabulary, which we we call the "axioms" of the theory.
Classical first-order logic shares its rules for how well-formed formulas look with some other logics, such as intuitionistic first-order logic. So if we have a first-order theory, we can -- at least in principle -- also consider which formulas can be derived from its axioms under the intuitionistic entailment rule. But in practice that rarely work well because when people design first-order theories to be used with the classical rules, they don't generally care to distinguish between formulations of the axioms that are classically equivalent but intuitionistically distinct. So the set of intuitionistic consequences of a theory that's made to be interpreted by classical logic can be somewhat random.