Short answers, pulled from the story.
Resolution in mathematical logic is a single inference rule that takes two clauses containing complementary literals, cancels those literals, and merges the remaining literals into a new clause called the resolvent. When combined with a complete search algorithm, it yields a sound and complete method for deciding whether a propositional formula is unsatisfiable.
Davis and Putnam introduced an early form of the resolution rule in 1960. John Alan Robinson made the decisive advance in 1965 by adding syntactical unification, which eliminated the combinatorial explosion in the original algorithm by instantiating variables only as far as needed during the proof.
The resolvent is the clause produced by applying the resolution rule to two input clauses. It contains all the literals from both clauses except the complementary pair that was resolved away. Robinson's definition of the rule also incorporated factoring, which unifies two literals within the same clause to ensure refutation completeness.
Refutation completeness means that if a set of clauses is genuinely unsatisfiable, the resolution procedure is guaranteed to derive the empty clause, confirming the contradiction. The empty clause, a clause with no literals, signals that the original conjecture follows from the axioms.
In first-order logic, the resolution rule is extended to use a most general unifier, a substitution that makes two expressions identical while committing as little as possible. Universal quantifiers are omitted as implicitly understood, and existentially quantified variables are replaced by Skolem functions before resolution is applied.
Paramodulation is a technique related to resolution that handles equality reasoning. It takes a clause containing a positive equality literal and replaces a unifiable subterm in a second clause with the other side of the equality, generating a new clause. Reflexive identities are excluded from the results.