Short answers, pulled from the story.
Logical consequence is the fundamental logical relationship in which a conclusion must be true whenever its premises are true. It is necessary and formal: it holds without regard to personal interpretations of the statements involved, and it can be determined a priori, without appeal to empirical evidence.
Alfred Tarski identified three features of an adequate account of logical consequence: it must rely on the logical form of the sentences, it must be a priori (determinable without empirical evidence), and it must have a modal component (involving necessity or impossibility). Tarski treated the a priori property as independent of formality.
Syntactic consequence holds when a conclusion can be derived from premises through formal proof rules within a system; its study is called proof theory. Semantic consequence holds when there is no model in which all premises are true and the conclusion is false; its study is called model theory. The turnstile symbol used for syntactic consequence was introduced by Frege in 1879, with its current usage dating to Rosser and Kleene in 1934-1935.
A formal consequence holds in virtue of logical structure alone and is valid for every possible substitution of content. A material consequence, by contrast, depends on the meanings of specific words. The inference "Fred is Mike's brother's son, therefore Fred is Mike's nephew" is material, not formal, because it relies on the meanings of "brother," "son," and "nephew."
A non-monotonic consequence relation is one where adding new premises to a set can defeat a conclusion that previously followed. The standard example: "Tweety can fly" follows from the premises that birds can typically fly and that Tweety is a bird, but this conclusion no longer holds once the premise "Tweety is a penguin" is added. Classical logical consequence is monotonic and cannot capture this kind of defeasible reasoning.
A warrant-based account holds that a good inference is one that never moves from justifiably assertible premises to a conclusion that is not justifiably assertible. This contrasts with truth-preservational accounts, which require only that true premises never lead to a false conclusion. The warrant-based view is associated with intuitionism.