What's the geometric intuition of the notions *reduced* and *nonreduced*? As well known, the notion reduce is defined by the notion nilpotent elements. For example, a scheme $\left(X, \mathcal{O}_{X}\right)$ is reduced if for every open set $U \subseteq X$, the ring $\mathcal{O}_{X}(U)$ has no nilpotent elements; $\left(X, \mathcal{O}_{X}\right)$ is reduced if and only if for every $P \in X,$ the local ring $\mathcal{O}_{X,P}$ has no nilpotent elements.
Question:  How to understand this notion? That is to say, can someone figure out a bit more about this, what it says, what it means, how to think of it geometrically, and so on. (Sorry for my rough expression, I am unable to express it precisely, however.) Furthermore, can any one give an example to make a distinction between the two concepts, reduced and unreduced.
 A: A commutative ring $R$ canonically maps to a product $\prod_P R/P$ of integral domains, where $P$ runs over all the prime ideals of $R$. The kernel of this map is the intersection $\cap P$ of all the prime ideals, which is the nilradical $\text{Nil}(R)$ (this is a classic exercise). So $R$ is reduced iff this map is injective iff $R$ embeds into a product of integral domains. Geometrically you can think of this condition as saying that elements of $R$ are determined by their "evaluations at prime ideals." This is one way to think about why "reduced" is one of the conditions defining a (not-necessarily-irreducible) variety.
The simplest example of a non-reduced ring to keep in mind is probably $k[x]/x^2$ where $k$ is, say, a field. The Spec of this ring arises naturally as a scheme-theoretic intersection; for example, it's the scheme-theoretic intersection of a circle and a line tangent to it (exercise). The fact that $\dim_k k[x]/x^2 = 2$ corresponds to the intuitive idea that this intersection has "multiplicity $2$," and this notion of multiplicity is one of the ingredients needed to make Bezout's theorem, that a plane curve of degree $d_1$ and a plane curve of degree $d_2$ intersect in $d_1 d_2$ points, true (there are two others, namely we need to work over an algebraically closed field and we need to consider intersections at infinity).
$\text{Spec } k[x]/x^2$ also naturally arises as the "walking tangent vector"; more formally, morphisms from this object to another $k$-scheme $X$ correspond to pairs consisting of a $k$-point $x \in X(k)$ and a Zariski tangent vector to $x$. So, for example, it can be used to define the Lie algebra of an algebraic group over $k$ (at least in characteristic $0$).
