Let $X$ be a scheme for which we have an appropriate moving lemma to define an intersection product of cycles. For example a smooth projective variety. Then if $V$ and $W$ are properly intersecting cycles in $X$, we can define an intersection product for properly intersecting cycles, $$ [V]\cdot [W] = \sum e_{Z} \cdot Z $$ with $$ e_{Z} = \sum_{i=0}^{\infty} (-1)^{i}\text{length}_{\mathcal{O}_{X, Z}} \operatorname{Tor}_{i}(\mathcal{O}_{V, Z}, \mathcal{O}_{W, Z}), $$ where the $Z$ are indexed over irreducible components of the set theoretic intersection $V \cap W$.
We can also define a scheme theoretic intersection via products. That is, we can define the scheme $V \times_{X} W$ which is a closed subscheme of $X$. Then we could take the associated cycle by defining $$ [V \cap W] = \sum n_{i} \cdot Z_{i} $$ where $n_{i} = \operatorname{length}_{ \mathcal{O}_{X, Z_{i}}} \mathcal{O}_{V \cap W, Z_{i}}$.
When do these agree? Do they ever? What is the motivation for defining it the former way when it seems like "intersection" is more naturally associated with pullback squares? And how does this generalise to general algebraic schemes? Can we define intersection theory there at all?