Search for the topic *vanishing theorems*. There are many such criteria, and perhaps this question should be community wiki?

Anyways, I'll highlight the one of the most common situations, for adjoint line bundles. Set $\omega_X = \Omega_X^{\dim X}$. Suppose that:

$E = \omega_X \otimes \text{ample}$

In this case you get the vanishing you want, which is called
the *Kodaira vanishing theorem*.

For example, if $X$ is Calabi-Yau or Fano, then $\omega_X$ is trivial or anti-ample respectively (by definition). In either case, for any ample $E$, the tensor product $E \otimes \omega_X^{-1}$ is also ample and so $E$ is of the form above, and so the vanishing you want holds.

Of course, many common classes of varieties are these types (there are varieties called log-Calabi-Yau and log-Fano which are also close enough). For example, toric varieties need not be Fano, but they are always ``close enough'' to give you the vanishing, they are log Fano.

If $X$ is not Calabi-Yau or Fano, then this sort of argument will fail, and you don't always get vanishing for arbitrary ample $E$. For example, if $X$ is a curve of genus $\geq 2$, then if you set $E = \omega_X$, then $E$ is ample. However, $H^1(X, E) = H^1(X, \omega_X) = H^0(X, O_X)$ which has dimension 1.

For vector bundles, again there are a lot of options when $E$ is various twists of $\Omega_X^j$. Look up Kodaira-Nakano-Akizuki vanishing.