Some additional comments to @diverietti answer,
Your question need more details, what do you mean of hermitian metric?, since the Griffiths positivity definition for smooth hermitian metric is different with continuous hermitian metric (which may not be smooth in general) or well defined singular hermitian metric
A continuous hermitian metric $h$ on a vector bundle $\pi : E \to X$ is said to be Griffiths positive, if there exists a smooth positive real (1, 1)-form $ω_X$ on $X$ such that
$$\pi^*\omega_X+\sqrt{-1}\partial\bar\partial \log h\leq 0$$
in the sense of currents
But a Hermitian vector bundle $E$ with smooth hermitian metric over a complex manifold $M$ to be Griffiths positive if $$\sqrt{-1}\Theta_{u\bar u}(X,\bar X) >0 \Longleftrightarrow R_{X\bar X u \bar u} >0$$ for any nonzero $(1,0)$ tangent vector $X$ of $M$ and nonvanishing section $u$ of the vector bundle $E$.
The good thing for Griffiths positivity of smooth hermitian metric is that $(E, h)$ is semi-positive if and only if its dual $(E^⋆, h^⋆)$ is semi-negative in the sense of Griffiths(I know it just for smooth hermitian metric and I don't know for continuous hermitian metric on vector bundle )(positivity duality property for other definitions like Nakano positivity is not known )
Ph. Griffiths in the following paper gave the following important conjecture for smooth hermitian metric
Ph. Griffiths, Periods of integrals on algebraic manifolds, III. Some global differential geometric properties of the period mapping, Inst. Hautes Etudes Sci. Publ. Math. ´ 38 (1970) 125–180
Conjecture: Let $E$ be an ample vector bundle, in the sense that $\mathcal O_E (1)$ is ample on $\mathbb P(E^⋆)$. Then $E$ admits a smooth Hermitian metric $h$ such that $(E, h)$ is positive in the sense of Griffiths
singular Hermitian metric on vector bundle is not well defined in general , but we know some result at least for some cases
A singular Hermitian vector bundle $(E, h)$ which is positively curved in the sense of Griffiths is weakly positive
We can extend Griffiths conjecture for singular hermitian metric that weak positivity of $E$ implies the Griffiths semi-positivity?
There are some good results for direct image of relative line bundle (note that direct image of relative line bundle is not line bundle in general and is vector bundle if we take its double dual to be reflexive )
See Paun survey paper