This question is motivated by this MO question, which seeks geometric meaning for irreducibility of an element. The first sentence is:
Consider a domain A and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory geometric interpretation of primeness.
I don't understand how to think of integral domains geometrically at all. What is the geometric intuition for primeness of an element?