First the nice case, then the general case.
Definition
Suppose \(Y\) is smooth, \(E\to Y\) is a vector bundle on \(Y\), and \(i:X = V(s)\hookrightarrow Y\) is a scheme embedded as the vanishing of some section \(s:Y\to E\). If \(s\) is a regular section, i.e. if \(X\) is also smooth, then \[c_{top}(E)\cap [Y] = [V(s)] = i_*[X].\] However, the left hand side doesn’t depend on \(s\). We call this the virtual fundamental class of \(X\), or rather, the class in \(A_*(X)\) which pushes forward to this class is the virtual fundamental class:
\begin{align*} i_*[X]^{vir} = c_{top}(E)\cap [Y]. \end{align*}