The general notion of a direct limit of a commuting system of embeddings, indexed by pairs in a directed set, has seen heavy use in set theory. It is the same notion as in category theory. I was surprised to find that the general definition does not appear in the book on model theory by Chang and Keisler (MR1059055). Who was the originator of this idea?
The definition of a direct limit of groups was given by Pontrjagin in his 1931 paper Über den algebraischen Inhalt topologischer Dualitätssätze
As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwarich in the other answer, the definition can be traced back at least to Lev Pontryagin.
For an early exposition in English, see
MR0007093 (4,84f). Lefschetz, Solomon. Algebraic Topology. American Mathematical Society Colloquium Publications, v. 27. American Mathematical Society, New York, 1942. vi+389 pp.
Specifically, direct limits are defined in Chapter 2, $\S$14 (p. 57).