Please forgive these very naïve remarks.  I am enjoying the chance to learn something about flatness in trying to contribute to this question.  First of all, since as pointed out here, the definition of flatness is that the object behaves as simply as possible under tensor product, it follows that the primary use of the concept is in applications of the tensor product.  Now there are three of these that come to mind, (after some review of the literature), namely
1)	forming fibers,
2)	localizing, (and changing rings)
3)	completions.

These are all local operations, looking at the inverse image of one point, restricting to a Zariski neighborhood of a point, and restriction to an analytic or formal neighborhood of one point.

Thus one wants to compare the geometry at a point with the geometry near that point.  E.g. given an algebraic subvariety through a point, one can take its algebraic germ, formal germ, or analytic germ, and then ask whether one can recover the original germ from these.  This is equivalent to asking whether one recovers the original ideal after extending to the localization or completion, and then restricting back to the original ring.

Flatness is the property that tells us yes to all these questions.  I.e. both the localization and the completion are flat over the original ring.  Moreover, the algebraic and analytic local rings form a “flat pair”, slightly stronger than one being flat over the other, and this apparently implies they have the same completions.  This lets us compare analytic and algebraic local rings, by comparing both with their completions. 

Reasoning of this sort allows Serre to prove geometric results such as those mentioned above as well as homological ones.  Homological results desired are of the sort that compare the analytic cohomology to the algebraic cohomology.  The simplest way to do this is to show that the analytic sheaves and their analytic cohomology is obtained from the algebraic ones by tensoring with flat objects, i.e. changing rings in the simplest way.  Then the desired results say that homomorphisms of sheaves, and cohomology of sheaves, commutes with this process of tensoring, i.e. of applying the functor of making these objects analytic.  Flatness is the key to all these results.

We have mentioned before the result that a surjective morphism of smooth varieties is flat if and only if the fibers have constant dimension.

Thus flatness of one object over another seems to imply a relation between their local geometric structures.  This is my take on Serre’s lovely paper GAGA, available free online at NUMDAM, after a brief perusal.  It certainly looks worth a careful read.

Another remark building on those above is that flatness is a natural weakening of the property of projectiveness, and hence for local rings, of freeness.  If  finite map behaves well locally when it defines a locally free module, what about a map that lowers dimension?  What is the closest thing to locally free that still holds when the fibers vary nicely but not smoothly?  I.e. a flat map is a slight weakening of a smooth map.  I am struggling here from ignorance.  Thanks for the question!