When is the product of two ideals equal to their intersection? Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds to the union $V(I\cap J)=V(I)\cup V(J)$. But a product $I.J$ gives a new subscheme $V(I.J)$ which has same support as the union but can be bigger in an infinitesimal sense. For example if $I=J$ you get a scheme $V(I^2)$ which is equal to "double" $V(I)$.
Vague Question  : What is geometric interpretation of $V(I.J)$ in general?
Precise question : When is $I\cap J=I.J$? Everybody knows the case $I+J=A$ but this is absolutely not necessary. For example if $A$ is UFD and $f,g$ are relatively prime then $(f).(g)=(f)\cap(g) $ but in general $(f)+(g)\neq A$ (e.g. $f=X, g=Y \in k[X, Y]$)
Thank you very much.
 A: A vague answer to the vague question:
When you want the union of $V(I)$ and $V(J)$ to behave well under deformations and to `count with multiplicity', then you may prefer to use the ideal $IJ$ rather than $I\cap J$. Let me give an example:
Take $V=V(x)$, $W=V(x-t)$ and $T=V(t)$ denote $V_0:=V\cap T=V(x,t)=W\cap T=:W_0$.
If you use intersection of ideals for the union of varieties you will get:
$(V\cup W)\cap T=V(x^2,t)$, and
$(V_0\cup W_0)\cap T=V(x,t)$.
While using product you will get:
$(V\cup W)\cap T=V(x^2,t)=(V_0\cup W_0)\cap T$.
A: Answer to the precise question: When $\mathrm{Tor}^1(A/I, A/J)=0$. 
Proof: We have the exact sequence
$$0 \to I \to A \to A/I \to 0$$
Tensoring with $A/J$, we get
$$0 \to \mathrm{Tor}^1(A/I, A/J) \to I/(I \cdot J) \to A/J \to A/(I+J) \to 0.$$
The left hand term is $0$ because $A$ is flat as an $A$-module.
Now, what is the kernel of $I \mapsto A/J$? Clearly, it is $I \cap J$. So the kernel of $I/(I \cdot J) \to A/J$ is  $(I \cap J)/(I \cdot J)$. We see that $I \cap J = I \cdot J$ if and only if $\mathrm{Tor}^1(A/I, A/J)=0$.
A: To add to David Speyer's answer, since this story continues with a rather interesting and illustrious history:
When $A$ is regular, the Tor functor satisfies the following property:

(1)  $\text{Tor}_1^A(M,N) = 0$ implies $\text{Tor}_i^A(M,N) = 0$ for $i>0$ for any two finitely generated modules.

(this is a theorem by Auslander in the geometric and unramified case and Lichtenbaum in the ramified case. (1) is called the rigidity of Tor).
It turns out that when $A$ is regular and local (so one can talk about depth), (1) implies

(2)  $\text{depth} (M) + \text{depth}(N) = \dim A + \text{depth} {M\otimes_AN}$

This stunning formula looks exactly the same as the property of "proper intersection" in intersection theory, except that one uses depth instead of dimension. Note that if $M=A/I, N=A/J$ then $M\otimes N = A/(I+J)$, which represents the intersection of $V(I)$ and $V(J)$, so this is very geometric.

(3) Talking about intersection theory, by Serre formula for intersection multiplicity, as all the Tors vanish, one can compute the intersection multiplicity of $V(I), V(J)$ by counting the length at the minimal components (i.e. the naive way). So you will have a generalization of Bezout theorem.

Finally, if $V(I)$ and $V(J)$ only intersect at isolated closed points, (2) implies (1) locally on the support of the intersection, so

(4) If $V(I) \cap V(J)= \{m_1, \cdots, m_n \}$ then $I\cap J = IJ$ if and only if $A/I, A/J$ are locally Cohen-Macaulay at the points $m_i$s.

You can find the last statement in Serre's Local Algebra book, V.6, Theorem 4, p 110 of the English version.
PS: Also, David did not mention his own interesting contribution, here.
A: I apologize in advance for resurrecting such an old question but I absolutely could not resist the urge of sharing a precise characterization for $I \cap J=IJ$, that I read recently in a beautiful short paper, when $I,J$ are monomial ideals in a polynomial ring.
First some terminology: Let $k$ be a field and $R=k[x_1,...,x_n]$ . Every monomial ideal $I$ of $R$ has a unique minimal monomial set of generators, usually denoted by $G(I)$ . For a set of monomials $T$ in $R$, let $\newcommand{\Supp}{\operatorname{Supp}}\Supp (T) :=\{i | x_i $ divides $m$ for some $m \in T \}$ .
With this, we can state the characterization: Let $I,J$ be monomial ideals in $k[x_1,...,x_n]$ , then $I \cap J=IJ$ if and only if $\Supp (G(I)) \cap \Supp (G(J))$ is empty.
This is Theorem 2.2 in https://link.springer.com/article/10.1007/s12044-019-0509-5
Here's the arxiv link https://arxiv.org/abs/1705.00488
