Use of indiscernibles in model theory
Some other classical uses of indisceribles due to Morley:
In the proof that $\kappa$-categorical theories are $\omega$-stable (for $\kappa\ge\aleph_1$), he constructs a model of size $\kappa$ realizing only countably many types over each countable set by taking a model generated by well ordered indiscernibles.
If for all $\alpha<\omega_1$ there is a model of size $\beth_\alpha$ omitting a type $p$, then there are arbitrarily large models omitting $p$, or, more generally, if an $L_{\omega_1,\omega}$ sentence has models of size $\beth_\alpha$ for all $\alpha<\omega_1$,then it has arbitrarily large models. These results need the Erd\"os-Rado partition theorem.
Eran,
As far as I know, indiscernibility is used in two ways in model theory. One, as you say, is to obtain many non-isomorphic models. This is for sure the classical use of indiscernibility.
Another, more modern one, is to have access to tools such as Ramsey's theorem and its uncountable version, the Erdős-Rado theorem. This is useful in some formulations of stability theory or (more recently, as in the work of Byunghan Kim) of simplicity. The point is that the notions of forking and dividing are cleaner to formulate in the presence of sufficiently indiscernible sequences. (So one typically works in large saturated structures in this context.) There are several modern references for stability, etc, where the use of indiscernibility is apparent, see for example Frank Wagner's "Simple theories", Mathematics and its applications, Kluwer Academic Publishers, 2000.
A third use of indiscernibility is fairly common in set theory, where it is the most common approach to defining the large cardinal notions known as sharps. A good reference for this use is Kanamori's "The higher infinite".
A further application of indiscernibles is to show that a consistent first-order theory with infinite models has models with many automorphisms. In particular, every first-order theory $T$ (in a countable vocabulary) possessing an infinite model has a model whose automorphism group has an undecidable first-order theory (in the vocabulary of groups). This result is due to Bludov, Giraudoux, Glass, and Sabbagh.
I believe this result can be extended to to show that every abstract elementary class $A$ which has members of unboundedly large cardinality also has members in every large enough power which possess undecidable automorphism groups. It follows that $A$ has non-rigid models. If $A$ also has rigid models in large enough cardinality, then $A$ is not categorical. This will hold for for classes defined by sentences or theories in some infinitary logics, e.g. $L_{\omega_{1} \omega}$, a most interesting case.
It may be very hard to construct rigid models in an AEC, and these are frequently not absolutely rigid, e.g. above the first $\omega$-Erdos cardinal, their rigidity can be destroyed by forcing. Under $GCH$ or $V=L$, one can neverthless attempt to build rigid models using diamonds $\lozenge_{\kappa}$ to eliminate automorphisms; in this way, the relative consistency of results refuting categoricity conjectures can be approached.