Example of the completion of a noetherian domain at a prime that is not a domain

Let $R=\mathbb{C}[x,y]/(y^2-x^2(x-1))$. This is the nodal cubic in the plane. Look at the prime $\mathfrak{p}=(x,y)$, corresponding to the nodal point. The completion here is isomorphic to $\mathbb{C}[[x,y]]/(xy)$.

It might be worth pointing out that you get an example of this by localizing at any point of a variety (scheme) which is irreducible but not (analytically/formally/étale) locally irreducible. In particular, any self-intersecting curve will give an example, just like the one above by Charles.

At least for algebraic local domains R, we have 1-1 correspondence between the minimal primes in the completion $\cap{R}$ and the maximal ideals in the integral closure of R in its quotient field. This will produce a lot of examples including curves near a point whose neighbourhood in the curve can't be covered by a single parameterization. In fact, various connections come to the fore here. For several ways to characterize the number of minimal primes in the completion of a local ring of an irreducible plane curve, one can look at S. S. Abhyankar's Chavounet prize winning paper "Historical Ramblings in Algebraic Geometry and related algebra".

Several connections are worth mentioning, eg. concepts such as Henselization and Hensel's lemma, Zariski's Main theorem (as a special case, it mentions that the completion of a normal algebraic local domain is again a normal domain), links associated to the singularities of algebraic curves, various reciprocity laws from Kummer to Artin in Algebraic Number Theory etc.