TITLE: Deformations and the dual numbers
QUESTION [10 upvotes]: The question I have is pretty straightforward, and its answer could very well be contained in some more complicated question(s) asked previously.  Here it is:
Why are deformations typically considered as structures over the ring of dual numbers?
My motivation is rather general: I just want to know more about deformation theory.  And this question seems like a place to start.  The problem for me is that all resources I have read just jump into the fray with "Consider the ring of dual numbers..." and define first-order deformations straight away.
I am sure that I am just missing something obvious; any nudge in the right direction will be greatly appreciated.

REPLY [6 votes]: Here are two main ideas which I found helpful when I was learning this stuff (very basic and handwavy, but good starting points to see where more precise and sophisticated statements are going):

see how, for instance, an algebra defined over $k[x]$ can be seen as a continuously-(in some sense)-varying $k$-indexed family of $k$-algebras.  In algebraical terms, you recover the elements of the family by localising at the ideal $x=t$ for each specific $t \in k$.  Geometrically, think of the Grothendieck-style picture where rings really represent some kind of nice spaces, but by duality, so the maps go the “wrong” way.  So a $k[x]$-algebra $R$ corresponds to a ring map $k[x] \to R$, and hence (since $k[x]$ represents the affine line $\mathbb{A}^1(k)$) to a map of spaces $X \to \mathbb{A}^1(k)$.  But then we can think of $X$ as the bundle of all its fibers, a family varying over $\mathbb{A}^1(k)$.
More generally, a widget defined over $k[x]$ can be thought of a a continuous (in some sense) map from $\mathbb{A}^1(k)$ into the space of widgets. 
now, see how just as $k[x]$ represented the affine line, $k[x]/(x^2)$ represents an “infinitessimally short line segment”, or a “walking tangent vector” (other answers include good sketches of what's meant by this and why it's a powerful viewpoint).  In particular, if $R$ correpsonds to a space $X$, a map $R \to k[x]/(x^2)$ corresponds to a map from the walking tangent vector into the space $X$, or in other words a point of $X$ and a tangent vector at that point.  Algebraically, we get a $k$-point of $R$ (by setting $x=0$), and (under good circumstances) a point of the Zariski tangent space.

So then putting these two together, one can see why a widget defined over the dual numbers should behave like some kind of an infinitessimal deformation: a widget, together with a tangent vector at that point in the space of widgets.  (This may be literally true if there's a good moduli space of widgets!)