Dear unknown, the most straightforward generalization of Bézout's theorem might be the following. Consider $\mathbb P^n $, projective space over the field $k$, and
$n$ hypersurfaces $H_1,...,H_n$in general position in the sense that their intersection is a finite set. Then, calling $h_i$ their local equations, Bézout says

$$\sum dim_k \mathcal O_{\mathbb P^n,P_i}/(h_1,...,h_n) =\prod deg (H_i) $$

The dimension on the left hand side is, of course, to be interpreted as the multiplicity with which to count the point $P_i$, seen as a fat point i.e. a zero-dimensional non-reduced scheme.

A related, more abstract point of view is the description of the Chow ring of $\mathbb P^n$ as $CH^\ast (\mathbb P^n)=\mathbb Z[x]/(x^{n+1})$ ( where $x$ is the class of a hyperplane in $\mathbb P^n$). From this point of view we have the following version of Bézout. Consider $r$ cycles $\alpha_1,...,\alpha_r$ on $\mathbb P^n$ with $\alpha_i \in CH^{d_i}(\mathbb P^n) $ and $d_1+...d_r \leq n$, . Then

$$deg \prod {\alpha_i} =\prod deg (\alpha_i)$$

the product of the $\alpha_i$'s on the left being calculated in the Chow ring and the degree $deg (\alpha) $ of a cycle $\alpha \in CH^d (\mathbb P^n)$ being the integer $t$ such that $\alpha =t . x \in CH^d(\mathbb P^n)=\mathbb Z .x $.

This is only the tip of the iceberg: a definitive answer would require a book. Fortunately that book exists and has been written, to our eternal gratitude, by Fulton: *Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer
Grenzgebiete. 3. Folge.
Springer-Verlag, Berlin, second edition, 1998.*