ac.commutative algebra - What does primary mean geometrically?

As Harry suggests in his answer, it is probably more intuitive to work with associated
primes, rather than the slightly older language of primary decompositions.

If $I$ is an ideal in $A$, an associated prime of $A/I$ is a prime ideal of $A$ which
is the full annihilator in $A$ of some element of $A/I$. A key fact is that for any
element $x$ of $A/I$, the annihilator of $x$ in $A$ is contained in an associated prime.

The associated primes are precisely the primes that contribute to the primary decomposition
of $I$. Geometrically, $wp$ is an associated prime of $A/I$ if there is a section
of the structure sheaf of Spec $A/I$ that is supported on the irreducible closed set $V(wp)$. E.g. in the example given in Cam's answer, the function $x^2 - x$ is not identically
zero on $X:=$ Spec ${mathbb C}[x,y]/(x y, x^3-x^2, x^2 y - xy),$ but it is annihilated
by $(x,y)$, and so is supported at the origin (if we restrict it to the complement of
$(0,0)$ in $X$ then it becomes zero).

The non-minimal primes of $I$ that play a role in the primary decomposition of $I$
(i.e. appear as associated primes of $A/I$) are the generic points of the so-called
embedded components of Spec $A/I$: they are irreducible closed subset of Spec $A/I$
that are not irreducible components, but which are the support of certain sections
of the structure sheaf.

An important point is that if $I$ is radical, so that $A/I$ is reduced, then
there are no embedded components: the only associated primes are the minimal primes
(for the primary decomposition of $I$ is then very simple, as noted in the question:
$I$ is just the intersection of its minimal primes).

There is a nice criterion for a Noetherian ring to be reduced: Noetherian $A$ is
reduced if and only if $A$ satisfies $R_0$ and $S_1$, i.e. is generically reduced,
and has no non-minimal associated primes. Geometrically, and applied to $A/I$
rather than $A$, this says that if $A/I$ is generically reduced, then the embedded
components are precisely the irreducible closed subsets of Spec $A/I$ over which
the nilpotent sections of the structure sheaf are supported. This may help
with your ``nilpotentification'' mental image.