Short answers, pulled from the story.
Non-monotonic logic is a formal logic whose entailment relation is not monotonic, meaning that adding new information can reduce the set of conclusions a reasoner can draw. Classical monotonic logics only ever accumulate conclusions; non-monotonic logics allow conclusions to be retracted when new evidence arrives.
A defeasible inference is a tentative conclusion that a reasoner draws based on available evidence, which can be retracted if further evidence contradicts it. Non-monotonic logics are designed specifically to capture and represent this kind of reasoning.
The wet grass example illustrates abductive reasoning: seeing wet grass leads a reasoner to conclude it rained, but that conclusion must be retracted if it is then learned that a sprinkler was running. Because the addition of new knowledge causes a conclusion to be withdrawn, any logic modelling this process is non-monotonic.
Autoepistemic logic is a non-monotonic logic designed for reasoning about knowledge. It handles situations where a formula stating that something is not known must be removed once that thing becomes known, which violates the condition of monotonicity.
Proof-theoretic formalization starts with non-monotonic rules of inference and uses fixed-point equations to define their application; default logic and autoepistemic logic are primary examples. Model-theoretic formalization instead restricts a monotonic logic to special models such as minimal models and derives non-monotonic rules from that restricted semantics, yielding paradox-free systems including first-order circumscription and the closed-world assumption.
Belief revision is the process of changing a set of beliefs to accommodate a new belief that may be inconsistent with the old ones. Because old beliefs must be retracted to maintain consistency when a new belief is added, any logic for belief revision is non-monotonic. This approach differs from paraconsistent logics, which tolerate inconsistency rather than removing it.