Why are flat morphisms "flat?"
A lot of people will tell you that flatness means "continuously varying fibres" in some sense, and that flatness was invented to have correspondingly nice consequences, which is true. But there is a way to expect this (vague) interpretation a priori from an alternative, equivalent definition:
An $A$-module $M$ is flat $\iff$ $I \otimes_A M \to IM$ is an isomorphism for every ideal $I$.
I would prefer to present this as the definition of flatness, and present the fact that tensoring with $M$ preserves exact sequences as a theorem. Why?
Thinking "geometrically", $I$ just corresponds (uniquely) to a closed subscheme $Z=Z(I)=$
$=Spec(A/I)\subseteq Spec(A)$. If we think of $M$ in the usual geometric way as a module of generalized functions on $X$ (like sections of a bundle), and $M/IM \simeq M\otimes_A A/I$ as its restriction to $Z$, then the above definition of flatness can be interepreted directly to mean that $M$ restricts nicely to closed subschemes $Z$.
More precisely, it says that what we lose in this restriction, the submodule $IM$ of elements which "vanish on $Z$", is easy to understand: it's just formal linear combinations of elements $i\otimes m$, with no surprise relations among them, i.e. the tensor product $I \otimes_A M$.
In topology, continuous functions "restrict nicely" to points and closed sets (by taking limits), so you can see, without much experience at all, how this definition corresponds in an intuitive way to continuity.
Having this motivation in place, the best thing to do is to check out examples along the lines of Dan Erman's answer to see the analogy with continuity and limits at work.
The key geometric meaning is that flat families are those families where the fibers vary "continuously". This notion allows one to talk about limits of families of algebraic varieties, which is particularly important in the study of deformation theory/moduli problems. Since the colloquial meaning of flatness also suggests a certain uniformity or lack of variation, one might imagine that this justifies its use in algebraic geometry.
For instance, if you have a flat family of projective varieties, then as Timo points out, the dimension of each fiber is the same. But more is true: the Hilbert polynomial of each fiber is also the same. This allows degeneration techniques. For instance, you can take a flat degeneration of your variety, compute a property about the degeneration, and then lift this information to your original variety.
I think that the geometric meaning of flatness is best understood via simple examples. Consider first $\text{Spec}(k[x,y,t]/(xy-t))\to \text{Spec}(k[t])$ via the natural map. This is a flat family. You can see this geometrically, as the fiber over t is a hyperbola when $t\ne 0$, and as $t$ approaches $0$, the hyperbola gets sharper and sharper and then it "breaks" into two lines when $t=0$.
Constrast this example with $\text{Spec}(k[x,y,t]/(txy-t))\to \text{Spec}(k[t])$. This is not a flat family. Here, when $t\ne 0$, the fiber is always the same hyperbola {xy-1=0}. But, when $t=0$, the fiber is an entire copy of $\text{Spec}(k[x,y])$. This pathological variation of the fibers is encoded by the fact that this is not a flat family.
I remember the following two quotes about flatness (I forgot who said/wrote this):
- For every geometric description of flatness there is a counterexample.
- Flatness is one of the few notions in algebraic geometry that were motivated by algebra and not by geometry.
This does not answer your question, I know... :)