Curvature of a connection (Ehresmann)

General fiber bundles

A connection on a fiber bundle is a particular kind of distribution, so we can think of the curvature of it. In this case, we have an isomorphism

being $V$ the vertical bundle of here, so the structure 1-form specified here takes value in the tangent spaces to the fibres.

If $v$ is the connection 1-form, then the curvature is the map

such that

where $P_H=id-v$ is the horizontal projection in $TE$.
I guess that, again can be seen like a vector bundle map

or. for $p\in E$:

Adapted from Wikipedia.

Principal $G$-bundles

If we are in the context of a connection not in a general bundle but in a $G$-principal bundle $P$, being $\mathfrak{g}$ the Lie algebra of the Lie group $G$, then the curvature is a $\mathfrak{g}$-valued 2-form on $P$:

and can be written in a simpler expression. Since in this case the connection 1-form $\omega$ can be seen as $\mathfrak{g}$-valued 1-form it turns out that

and the curvature of the connection is defined by

for certain bracket defined in Lie algebra-valued differential forms.

The Riemann curvature tensor is a special case of curvature of a connection on a principal bundle.

Vector bundles

(see Wikipedia )
Consider a vector bundle $E\to M$. In this case the connection $D$ would be a vector bundle connection and the connection 1-form can be expressed in local frames with the connection 1-forms $\mathrm{\Theta }$, or better said, a $\mathfrak{gl}(n)$-valued 1-form. Then the curvature is

or, in components,

called the curvature 2-forms.

Indeed, we also call the curvature of $D$ to the operator (see @baez1994gauge page 243)

where $v,w$ are tangent vectors of $M$ and $s$ is a section of $E$. Given a local frame $e_i$, the connection can be expressed in terms of the vector potential or Christoffel 1-form $A_{\mu j}^i$ or ${\mathrm{\Theta }}_{\mu j}^i$, depending on the reference. So the curvature is expressed as

which is another way to express equation $(c)$.

If the vector bundle is the tangent bundle and the connection is the Levi-Civita connection of a metric then this curvature is the Riemann curvature tensor of the metric by the Cartan's second structural equation.

See also the note Gaussian curvature#Relation to the curvature of a connection.

Relation to holonomy

@baez1994gauge page 247.
Regarding the holonomy of the connection, still in the context of a vector bundle, it turns out that if we parallel transport a vector $v\in E_p$ around a little square $\gamma$ of side $ϵ$ in the coordinates $x_{\mu },x_{\nu }$, the result is a vector $v^{\prime }\in E_p$ such that

or in other words

See @needham2021visual page 290 for an explanation, in the context of surfaces and tangent bundles, why curvature is related to holonomy.