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### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

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 ...

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### Factoriality of complete intersections

Let $X\subset\mathbb{P} _{\mathbb{C}}^N$ be a normal complete intersection.
$X$ is called factorial if every Weil divisor on it is Cartier;
equivalently if all local rings $\mathcal{O}_{X,x}$ are ...