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.

For the latter question, the most classical theorem (of Petri, I think) is that any smooth non-hyperelliptic curve 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 by Voisin.

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