When is an intersection of varieties finite

Question

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ finite, then $|V| \le \prod d_i$. My question is when do $p_i$ guarantee that $V$ is finite (and then Bezout bounds how many points are in V)? I am looking more for a geometric criterion (such as something to do with the tangent spaces) rather than an algebraic criterion. Reading up on this, I see that the criterion of transversal intersection is related to this, but I do not understand too well. Is there an easier criterion for $|V|$ finite when all the degrees are equal (this is easy when $d=1$, as these are hyperplanes. Is $d=2$ significantly harder?)