On the Wikipedia page about algebraic varieties http://en.wikipedia.org/wiki/Algebraic_variety, a sentence reads as follows:

[[A more significant modification is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in coordinate rings aren't seen as coordinate functions.

From the categorical point of view, nilpotents must be allowed, in order to have finite limits of varieties (to get fiber products).]]

So I am wondering if there is an intuitive example to get non-reduced 'schemes' from reduced 'algebraic varieties' (probably by taking fiber product or alike)?