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A relation R is well-founded if every non-empty subset has an R-minimal element. Well-founded relations support recursion and induction: any function can be defined recursively by specifying its value on R-minimal elements and then using values at 'R-smaller' arguments. This generalizes finite induction to potentially infinite domains.
From your work with binary relations, you know that a relation R on a set A is just a collection of ordered pairs — saying "x R y" means x is related to y in a specific direction. Well-foundedness adds a structural constraint on how those pairs are arranged: a relation is well-founded if there are no infinite descending chains. More precisely, every non-empty subset S of the domain must contain an R-minimal element — some element m ∈ S such that nothing in S is R-smaller than m. The natural numbers under < are the canonical example: every non-empty set of natural numbers has a least element. What fails in the integers under < is exactly well-foundedness — the negative integers form an infinite descending chain with no minimum.
Why does well-foundedness matter? Because it is precisely the structural property that makes recursion and induction work. In finite induction, you prove P(0), then prove P(n) → P(n+1), and the well-foundedness of < on ℕ guarantees the argument reaches every natural number. Well-founded induction generalizes this: to prove a property P holds for all elements of a well-founded relation, prove that for any x, if P holds for every R-smaller element, then P holds for x. The well-foundedness condition ensures you never get trapped in an infinite regress of "but what about something smaller?"
Transfinite recursion is the corresponding definition principle. If you want to define a function f on a well-founded domain, you specify: given the values of f on all R-predecessors of x, what is f(x)? Well-foundedness guarantees that this specification uniquely determines f everywhere, because the "look back at smaller values" process always terminates at R-minimal elements, which serve as base cases. Recursive definitions on trees, for example, work because trees are well-founded under the "child of" relation — you always bottom out at leaves.
The generalization to infinite structures beyond ℕ is what makes this concept central to set theory. Ordinal numbers, which you will study next, are essentially the canonical well-ordered sets — totally ordered well-founded structures — and transfinite recursion over them is how set theorists construct the cumulative hierarchy, define operations on infinite cardinals, and reason about the structure of the mathematical universe itself. Well-foundedness is the bridge between finite induction you already know and the transfinite reasoning set theory requires.