In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are commutative — more generally, that homotopy groups of H-spaces are commutative, in particular, higher homotopy groups are commutative.
This argument takes its name from its appearance in a 1962 Eckmann-Hilton paper (Group-like structures in general categories, I. Multiplications and comultiplications, Mathematische Annalen, June 1962, Volume 145, Issue 3, pp 227–255) but stems, it seems, from previous work from the same authors (Homotopy and duality, 1959).
It also appears explicitly in algebraic geometry, in Grothendieck's 1961 SGA 1 seminar Revêtements étales et groupe fondamental, devoted to the theory of the fundamental group of schemes. Namely, in Exposé XI (Examples and complements), Section 2 (Abelian varieties), Grothendieck writes (bottom of page 287 of the Springer-Verlag edition) : “On the other hand, since the functor $X\mapsto \pi_1(X)$ from pointed schemes $X$ to groups commutes with product, it transforms a group in the first category to a group in the category of groups, i.e., a commutative group.”
Does anybody know whether the two groups of people were aware of one another? Was this categorical argument already clear at that time ?