As Pete already indicated, Mumford's theorem says that for any projective variety $X\subset \mathbb P^n$, its Veronese emberdding $v_d(X)\subset \mathbb P^N$ is cut out by quadrics, for $d\gg0$. So a reasonable question is for the variety with a fixed projective embedding (such as Grassmannians in the Plucker embedding and not in some other random embedding).

For the latter question, the most classical theorem (by some classical Italian mathematician, Petri, I think) is that any smooth non-hyperelliptic curve, in its canonical embedding, is cut out by quadrics.

There is a vast generatization of this property: $X\subset \mathbb P^n$ satisfies property $N_p$ if the first syzygy of its homogeneous ideal $I_X$ is a direct sum of $\mathcal O(2)$, the second syzygy is a direct sum of $\mathcal O(3)$, etc., the $p$-th syzygy is a direct sum of $\mathcal O(p+1)$. In this language, $X$ is cut out by quadrics is equivalent to the property $N_1$.

A 1984 Green's conjecture is that a smooth nonhyperelliptic curve satisfies $N_p$ for $p=$ its Clifford index minus 1. I think it has been proved for generic curves of any genus by Voisin.

Another notable case: the ideal of $2\times 2$ minors of a $p\times q$ matrix has property $N_{p+q-3}$.