Let $R$ be a local Noetherian ring. What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric interpretation?


Regular rings have clear and famous geometric interpretation: They correspond to non-singular (affine) varieties.
A (local) complete intersection rings can be thought of as a local ring at a point, such that at this point the variety can be defined by minimal number of relations, i.e. a regular sequence. In more geometric flavor a local complete intersection subscheme is the one whose normal bundle is still a vector bundle (a property similar to the non-singular case) but it can be singular.
Gorenstein rings have more complicated geometric meaning: They are local rings of affine varieties which may be singular at the point, but the maximal $R$-regular sequence in the maximal ideal i.e. the associated complete intersection subscheme is an irreducible space.
In general Gorenstein and complete intersection (and Cohen-Macaulay) local rings are substitutes of the regular local rings. It means that they might be singular, but their singularities is as nice as it can be. So we have the following strict inclusions:

Regular rings $\subset$ CI rings $\subset$ Gorenstein rings

The simplest example of a complete intersection ring which is not regular is the ring $k[x]/(x^{2})$. The maximal ideal is the ideal $(\overline{x})$ and of course $x$ is a regular element in $k[x]$. For a ring which is Gorenstein but not CI, consider the $0$-dimensional local ring $k[x,y]$$/(x^{2},y^{2},z^{2}$$,xz,yz, z^{2}-xy)$. The unique maximal ideal here is the ideal $(x,y,z)$ but you can see that the ideal $(x^{2},y^{2}, xz,yz, z^{2}-xy)_{(\overline{x}, \overline{y}, \overline{z})}$ is not generated by a regular sequence.

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    $\begingroup$ Are there textbooks on this subject which do not provide examples to show that the inclusions are strict? $\endgroup$ – Mariano Suárez-Álvarez Mar 6 '14 at 20:09
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    $\begingroup$ Mariano Suárez-Alvarez! Note:"explain how you reached to that examples". I don't want an example. I want to know how to construct. $\endgroup$ – user 1 Mar 6 '14 at 20:15
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    $\begingroup$ Of course in general it is hard to give a general recipe of how to construct such examples. In special cases there are some results which can be helpful. For example a ring of dimension $0$ is complete intersection iff the Fitting ideal of its maximal ideal is non-zero. Obviously this gives a possibility to construct singular rings with CI. $\endgroup$ – Darius Math Mar 6 '14 at 21:21

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