Given a finite-dimensional semisimple complex Lie algebra $\mathfrak{g}$, the Bernstein-Gelfand-Gelfand category $\mathcal O$ is the full subcategory of $\mathfrak g$-modules satisfying some finiteness conditions. It contains all finite-dimensional modules as well as all highest-weight modules, it's Noetherian and Artinian, and it's Abelian. It's clear to me why you would want to work in some full subcategory of $\mathfrak g$-modules which has the above properties, but why $\mathcal O$? Is it minimal in some sense with respect to these properties and/or some other important properties? 



Why the BGG category O?
I'd add to what Ben says the observation that people have found a number of different module categories valuable for different purposes within this same Lie algebra context. (And some Lie theory people don't really find category $\mathcal{O}$ to be all that important in their own work.) Even two people close to the original construction, Joseph Bernstein and Sergei Gelfand, found it more useful to broaden the study to categories satisfying somewhat different finiteness conditions in their further work on projective functors. 


Much of the original motivation for category $\mathcal{O}$ came from a rethinking of the classical finite dimensional theory combined with an attempt to understand better the problems raised by Verma's thesis and later work by Jantzen. Here is where the translation functors really come into their own, along with the refined use of central characters and blocks. But the "correct" module category to study depends on which problems are being studied. The category of all modules for a universal enveloping algebra is definitely too big for practical purposes, but within it there are many attractive subcategories.


P.S. As these answers illustrate, there can be several different answers to the "why' question asked. The answers have certainly evolved over time, as illustrated in part by the series of BGG and BG papers. For example, the BGG category turns out to be just right for BGG Reciprocity, due to the special nature of projective objects in this category relative to Verma modules. (On the other hand, the earlier prototype of BGG Reciprocity in prime characteristic involved just finite dimensional modules and therefore required just the natural category of such modules for the finite dimensional restricted enveloping algebra of the Lie algebra of a semisimple algebraic group.) 


