A lattice $(L,\leq)$ is said to be modular when $$ (\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b, $$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.)
The ideals of a ring form a modular lattice. So do submodules of a module. These facts are easy to prove, but I have never seen any striking examples of their utility. Actually, in a seminar I took part in, the speaker said the modularity condition wasn't very natural and that there was an ongoing search for better ones (this was in the context of the Gabriel dimension and its generalization to lattices -- unfortunately, I didn't understand much of that).
I would like to see some motivation for this notion. That is, I would like to know when it is useful, and if it is natural. At the moment, it doesn't look any more natural to me than any random condition in the language of lattices. If you could shed some light on the opinion I quote in the previous paragraph, it would be very helpful as well. I would be especially interested in algebraic motivation, as I know very little about other areas if mathematics.