Such first-order infinitesimal deformations allow one to compute the Zariski tangent space when good moduli space exists. You can find some nice motivational remarks in the first chapter of Hartshorne's Deformation Theory. See esp. the two paragraphs preceding the Exercises 1.1.
You may find it helpful to first understand analogous applications of dual numbers in simpler contexts. Below is an extract from one of my old sci.math posts which has some references.
What is the factor ring R[x]/(x^2) ?
It is known as the algebra of dual numbers over R, for R a commutative ring. It and its higher order analogs R[x]/(x^n) prove useful when studying derivations. E.g. they permit easy transfer of properties of homomorphisms to derivations -- see section 8.15 in Jacobson, Basic Algebra II. They yield algebraic models of tangent spaces.
They've been applied in many contexts, e.g. deformation theory [2], numerical analysis [3] (along with Levi-Civita fields [4]), where they're viewed simply as truncated Taylor / power series, and in Synthetic Differential Geometry [1], another rigorization of inifinitesimals based on work of Lawvere and Kock. SDG employs nilpotent infinitesimals, unlike Abe Robinson's nonstandard analysis which has invertible infinitesimals, hence infinities.
[1] Bell, J. L. Infinitesimals. Synthese 75 (1988) #3, 285--315.
[2] Szendroi, B. The unbearable lightness of deformation theory,
a tutorial introduction.
http://www.maths.warwick.ac.uk/~balazs/defth.ps
[2] M. Berz, Differential Algebraic Techniques,
in "Handbook of Accelerator
Physics and Engineering, M. Tigner, A.Chao (Eds.)" (World Scientific, 1998)
http://bt.pa.msu.edu/cgi-bin/display.pl?name=dahape
http://bt.pa.msu.edu/NA/
http://bt.pa.msu.edu/pub/papers/