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 understand analogous application of dual numbers in simpler contexts first. 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 it's higher order analogs  R[x]/(x^n)  prove useful when studying 
derivations. E.g. they permit 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/ 

[4] http://bt.pa.msu.edu/pub/papers