Here is an important application. It is probably in the relevant chapter of Voisins book.
Let $V^k \subset CP^n$ be a smooth projective variety, $d$ a number and $s$ be a section of the $d$th power of the Hopf bundle on $CP^n$. Assume that $s$ is transverse to the zero section and that $V$ and $s^{-1}(0)$ are transverse. Let $W$ be the intersection. Then $\pi_i(W) \to \pi_i(V)$ is an iso if $i< k$. If $d$ were $1$, then this is just the hyperplane theorem. For larger $d$, the sections of $H^{\otimes d}$ define an embedding $f$ into a higher-dimensional projective space $P^{N(d)}$ (the Veronese embedding, see Georges answer) and the section $s$ defines a hyperplane in that larger space. Then apply the hyperpalane theorem to $f(W) \to f(V)$; and you get the result.
A complete intersection variety $V^k \subset P^n$ is the common zero set of $n-k$ sections $s_1, \ldots,s_{n-k}$; $s_i$ a section in $H^{\otimes d_i}$ and these sections are in general position. By an iterative application you get that $\pi_i (V)=\pi_i(P^n)$ for $i< k$. Hence projective tori of dimension $>1$ are not complete intersections.