Steven already explained a bit about the key to the answer: Synthetic differential geometry (cf. nLab where the hints on a higher categorical analogue are also present), but I would like to put it in a much broader perspective, though more in words and references than really explaining, mainly due to space, time and expertise limits.

While there are mentions, in several of the answers above, of the (possibly relative) module of Kähler differentials leading to the algebraic version of cotangent space used in algebraic and analytic geometry; this partial notion predates a little bit the more fundamental work of Grothendieck, who invented an inherent geometrical way to found a differential calculus in geometry. Similarly to the differentiation in topological vector spaces, the basic idea is to approximate the maps with linear maps, but this time Grothendieck considered maps among sheaves of $\mathcal{O}$-modules over schemes; he described the linearization in the language of operations on sheaves in terms of infinitesimal neighborhoods of the diagonal $\Delta\subset X\times X$; these are described in terms of nilpotent elements in the structure sheaf; one can also define the related notion of infinitesimally close generalized points; the infinitesimal neighborhoods build up an increasing filtration, which induces a dual filtration on the hom-spaces, so called differential filtration. The union of the differential filtration is the differential part of the hom-bimodule, and its elements are regular differential operators. A "crystaline" variant of the picture related to divided powers leads to appropriate treatment of differential calculus in positive characteristics. The notion of crystal of quasicoherent sheaves is bases on the notion of infinitesimally closed generalized points; the geometric picture with pullbacks of sheaves, leads to a definition as sort of descent data, cf.

- P. Berthelot, A. Ogus,
*Notes on crystalline cohomology*, Princeton Univ. Press 1978. vi+243.
- J. Lurie,
*Notes on crystals and algebraic $D$-modules*, in Gaitsgory's seminar, pdf

These are a dual point of view on D-modules. The descent data in abelian context are equivalent to certain formally defined connection operator, called Grothendieck connection in this case. There are now abstract versions of the algebraic correspondence between descent data in abelian context and flat Kozsul connections for the associated "Amitsur" complexes (work of Roiter, T. Brzeziński and others, cf. connection for a coring). On the other hand, Grothendieck immediately came up with nonlinear version of crystals (crystals of schemes) which are dual point of view on what some now call D-schemes.

Grothendieck's point of view on differential calculus has been soon after the discovery at the end of 1950s, introduced in works of Malgrange, Kodaira and Spencer in the development of obstruction and deformation theory for differential equations. Both works together in late 1960s motivated Lawvere, Kock and Dubuc to extend that geoemtric approach to differential calculus into differential geometry. Dubuc introduced $C^\infty$-schemes as yet another approach to manifolds, in the spirit of the theory of schemes. Lawvere did not look only at then recent work of Grothendieck (and Malgrange, Spencer...), but also at classical work on "synthetic geometry". This is a terminology which requires caution: in 19th and early 20th century, synthetic was viewed as differing from coordinatized, analytic, and pertained to either work from axioms, not referring to coordinate and even metric aspects, and some people in axiomatic descriptive geometry refer to their geometry as synthetic even now in that "clean", but less powerful sense. Another sense is that it is close to the engineering point of view that the path of a particle can be considered either as a point in the space of paths or as a map from interval into the space, what implies that the infinite-dimensional spaces of paths should exist and one should have the exponential law, i.e. we need to embed our category of spaces into closed monoidal category; there are many such embeddings of the category of manifolds available now, and some models of them offer the model $D$ of infinitesimals, which represents the functor of taking the tangent space in particular. This model has been shaped with having in mind the Grothendieck's field of dual numbers in algebraic geometry, but the language and multiplicity of models made it very flexible in the approach of the *synthetic differential geometry* of Kock and Lawvere. First of all they had an independent axiomatic approach as well as study of the topos theoretic models; including the study of Cahiers topos which is even more faithful to the Grothendieck's point of view. In all these models, they had nilpotent infinitesimals, like in scheme theory, but not like infinitesimals in nonstandard analysis. More recent approach of Moerdijk-Reyes offered both nilpotent and non-nilpotent infinitesimals (though possible variant related to nonstandard analysis is not known to me). For a differential geometer there are many attractive tools in synthetic differential geometry like infinitesimal simplices, enabling intuitive and effective of many quantities involving differential forms and geometry.

On the other hand, the usual differential geometry is faithfully embedded into synthetic models, so one is bound to be conservative, i.e. not to get results about usual notions in manifolds theory which are inconsistent with the usual definitions. One just gets more intuitive and technical power.

I should also mention that the Grothendieck's picture with the infinitesimal thickenings, aka resolutions of diagonals, leading to differential calculus, can be extended to noncommutative spaces rerpesented by abelian categories "of quasicoherent modules". This has been done in 1996 preprints

- V. A. Lunts, A. L. Rosenberg,
*Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings*, MPI 1996-53 pdf, *II. D-Calculus in the braided case. The localization of quantized enveloping algebras*, MPI 1996-76 pdf

The resulting definition of the rings of regular differential operators on noncommutative rings has been used in the study toward the Beilinson-Bernstein correspondence for quantum groups in later published two articles, which however skip the geometric derivation of the definition of the differential operators used:

- V. A. Lunts, A. L. Rosenberg,
*Differential operators on noncommutative rings*, Selecta Math. (N.S.) **3** (1997), no. 3, 335--359 (doi); sequel: *Localization for quantum groups*, Selecta Math. (N.S.) **5** (1999), no. 1, pp. 123--159 (doi).

Somewhat similar analysis in infinite-categorical setup of a $(\infty,1)$-version of the Cahiers Topos is in recent master diploma

- Herman Stel,
*∞-Stacks and their Function Algebras – with applications to ∞-Lie theory*, Utrecht 2010, webpage, pdf

under the guidance of Urs Schreiber. This work leads to a correct theory of higher Lie algebroids.