on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
Let $\mathcal{C}$ be a (cofibrantly generated) model category and let $\mathcal{D}$ be any category, regarded as a diagram-shape in the following.
Write $[\mathcal{D}, \mathcal{C}]_{proj}$ for the projective model structure on the functor category of functors from $\mathcal{D}$ to $\mathcal{C}$, hence of $\mathcal{D}$-diagrams in $\mathcal{C}$.
A functor/diagram $X : \mathcal{D} \to \mathcal{C}$ is a projectively cofibrant diagram in $\mathcal{C}$ if it is a cofibrant object in the projective model structure $[\mathcal{D}, \mathcal{C}]_{proj}$.
We unwind the condition in def. 1.
First of all it says of course that a diagram $X \colon \mathcal{D}\to \mathcal{C}$ is projectively cofibrant precisely if the inclusion $\emptyset \to X$ of the initial diagram has the left lifting property with respect to natural transformations of diagrams
which are projective acyclic fibrations, hence which are such that for each $c \in \mathcal{C}$ the component $\eta_c : A(c) \to B(c)$ is an acyclic fibration in $\mathcal{C}$.
This in turn means that $F$ is projectively cofibrant precisely if for every diagram of natural transformations
with $p$ as above, there exists a lift $\sigma$ in
making the triangle commute.
The main point of projectively cofibrant diagrams is that the ordinary colimit over them is a presentation of the homotopy colimit:
because the (colimit $\dashv$ constant diagram)-adjunction
is a Quillen adjunction (because $const$ is by the very definition of the projective model structure a right Quillen functor), the homotopy colimit, being the left derived functor $\mathbb{L}\underset{\longrightarrow}{\lim}$ of the colimit, is computed as the ordinary colimit evaluated on a cofibrant resolution $Q X$ of a diagram $X : \mathcal{D} \to \mathcal{C}$:
A span diagram $X_1 \leftarrow X_0 \to X_1$ is projectively cofibrant precisely if the two morphisms are cofibrations in $\mathcal{D}$ and $X_0$, hence all three objects, are cofibrant.
The colimit over such a diagram is the homotopy pushout of the span.
A cotower diagram
is projectively cofibrant precisely if every morphism is a cofibration and if the first object $X_0$, and hence all objects, are cofibrant in $\mathcal{D}$.
The colimit over such a diagram is a homotopy sequential colimit.
A parallel morphisms diagram
is projectively cofibrant precisely if $X_0$ is cofibrant, and if the morphism $(f,g) : X_0 \coprod X_0 \to X_1$ is a cofibration.
This implies that also $f$ and $g$ are cofibrations and hence that $X_1$ is cofibrant.
The colimit over such a diagram is a homotopy coequalizer.
Let $\mathcal{C} =$ sSet${}_{Quillen}$ be the standard model structure on simplicial sets. Then $[\mathcal{D}, \mathcal{C}]_{proj}$ is the projective model structure on simplicial presheaves.
For the following see at model structure on simplicial presheaves the section Cofibrant objects for more details (due to Dan Dugger).
A sufficient condition for a diagram $X : \mathcal{D} \to sSet$ to be projectively cofibrant is:
$X$ is degreewise a coproducts of representables
the degenerate cells in each degree form a separate coproduct summand;
A split hypercover is of this form.
For $X : \mathcal{D} \to sSet$ any simplicial presheaf, a cofibrant resolution is given by
where the coproduct runs over all sequences of morphisms between representables $U_i$, as indicated.
See the references at homotopy colimit and generally at model category.
Related discussion is at
Related discussion is for instance also in
where cofibrant cotowers are mentioned as example 2.3.15.