As mentioned in the comments, the *(pseudo)effective cone* $\overline{\mathrm{Eff}}(X)$, defined as the closure of the cone of all effective divisors on $X$, is certainly an object of study, and Lazarsfeld's book is a good reference. Your complaint that he doesn't say much about its structure is surely related to the fact that so little is known! Here are a few general things I'm aware of:

The interior of the effective cone is the *big cone*, i.e., the cone of line bundles with positive volume.

The dual of the effective cone is the *cone of moveable curves*, see Boucksom-Demailly-Paun-Peternell.

As part of their work on the minimal model program, Birkar-Cascini-Hacon-McKernan prove that log Fano varieties have finitely generated effective cones.

And here are a couple specific instances where one knows more:

When $X$ admits an action by a solvable group with a dense orbit, the effective cone is generated by the components of the complement of the orbit. (This works when $X$ is, e.g., a toric variety or a Schubert variety.)

There's been a lot of recent work on the case $X=\overline{M}_{0,n}$, see e.g., Hu-Keel, Hassett-Tschinkel, Castravet-Tevelev.