Consider the union of the axes $\frac{\Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin.
Now consider the nodal cubic $\frac{\Bbbk[x,y]}{(y^2-x^2(x+1))}$. There are two "branches" passing through the node, but they are not seen by the local ring at the origin, which is a domain (as the localization of a domain).
In this comment, Roland explains this issue is resolved by passing to the (strict) Henselization of the local ring at the origin, which does have two minimal primes.
How to calculate the (strict) Henselization of the local ring of the nodal cubic at the node? (How to calculate Henselizations in general?)
Thinking of (strict) Henselizations as universal covers, why intuitively does the (strict) Henselization of the node acquire the "missing" minimal prime?