Note that this differs from the definition of the well-order relation, where we do not require an R-minimal element but R-least element instead.
A set equipped with a well-founded relation is sometimes said to be a well-founded set. A well-founded set is a partially ordered set which contains no infinite descending chains, or equivalently, a partially ordered set in which every non-empty subset has a minimal element. If the order is a total order then the set is called a well-ordered set.
One reason that well-founded sets are interesting is because a version of transfinite induction can be used on them: if (X, <=) is a well-founded set and P(x) is some property of elements of X and you want to show that P(x) holds for all elements of X, you can proceed as follows:
A familiar example of a well-founded relation is the ordinary < relation on the set of natural numbers N. Every non-empty subset of the natural numbers contains a smallest element. This is known as the well-ordering principle.
Well-foundedness is interesting because the powerful technique of induction can be used to prove things about members of well-founded sets. For the example of the natural numbers above, this technique is called mathematical induction. When the well-founded set is the set of all ordinal numbers, and the well-founded relation is set membership, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. See the articles under those heads for more details.