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.
For reference:
Modularity: $x\leq b \implies x\vee(a\wedge b)=(x\vee a)\wedge b$
Algebraically it is a relaxed distributivity condition for the meet and join operations. Graphically, it means that the forbidden Pentagon diagram will have one or more of its sides crushed. I can't think of any more proof for their naturality other than the fact the submodules of a module and the set of normal subgroups of a group are all modular lattices. Groups and modules are very natural!
Similarly its cousin the distributive lattice is algebraically distributivity of meet and join over each other. Modules do not normally have a distributive lattice of submodules.
Distributive lattices are natural because their prototype is the lattice of subsets of a given set with intersection and union operations. It is known that every distributive lattice is lattice isomorphic to such a set lattice.
P.S.: I didn't know this before, but I found that von Neumann apparently made use of complemented modular lattices in his book Continuous Geometry, so I would also look there for inspiration.
Added: You were requesting some places where modularity was explicitly used. When Ward and Dilworth went about abstracting the study of ideals in a ring to "multiplicative lattices", they managed to do primary decomposition in what they called Noether lattices. These were of course supposed to generalize the lattice of ideals of a Noetherian ring, and general multiplicative lattices are far too wild, so they needed to make some additional natural requirements for the multiplicative lattice. Among these assumptions were the ACC (to make it Noetherian), the property that every element should be a join of principal elements, and finally modularity of the lattice. I'm not an expert in the topic but I think modularity was probably crucial in their proofs using residuals.