The notion of weighted limit is naturally understood from the point of view on limits as described at representable functor.
Weighted limits make sense and are considered in the general context of $V$-enriched category theory, but restrict attention to $V=$ Set for the moment, in order to motivate the concept.
Let $K$ denote the small category which indexes diagrams over which we want to consider limits and eventually weighted limits. Notice that for
a Set-valued functor on $K$, the limit of $F$ is canonically identified simply with the set of cones with tip the singleton set $pt = \{\bullet\}$:
This means, more generally, that for
a functor with values in an arbitrary category $C$, the object-wise limit of the functor $F$ under the Yoneda embedding
which appears in the discussion in example 1 at representable functor can be expressed by the right side of
(Recall that this is the limit over the diagram $C(-,F(-)) : K \to Set^{C^{op}}$ which, if representable defines the desired limit of $F$.)
The idea of weighted limits is to
allow in the formula above the particular functor $\Delta pt$ to be replaced by any other functor $W : K \to Set$;
to generalize everything straightforwardly from the Set-enriched context to arbitrary $V$-enriched contexts.
The idea is that the weight $W : K \to V$ encodes the way in which one generalizes the concept of a cone over a diagram $F$ (that is, something with just a tip from which morphisms are emanating down to $F$) to a more intricate structure over the diagram $F$. For instance in the application to homotopy limits discussed below with $V$ set to SimpSet the weight is such that it ensures that not only 1-morphisms are emanating from the tip, but that any triangle formed by these is filled by a 2-cell, every tetrahedron by a 3-cell, etc.
Let $V$ be a closed symmetric monoidal category. All categories in the following are $V$-enriched categories, all functors are $V$-functors.
A weighted limit over a functor
with respect to a weight or indexing type functor
is, if it exists, the object $lim^W F \in C$ which represents the functor (in $c \in C$)
i.e. such that for all objects $c \in C$ there is an isomorphism
natural in $c$.
(Here $[K,V]$ is the $V$-enriched functor category, as usual.)
In particular, if $C = V$ itself, then we get the direct formula
This follows from the above by the coend manipulation
For $V$ some category of higher structures, the local definition of homotopy limit over a diagram $F : K \to C$ replaces the ordinary notion of cone over $F$ by a higher cone in which all triangles of 1-morphisms are filled by 2-cells, all tetrahedra by 3-cells, etc.
One can convince oneself that for the choice of SimpSet for $V$ this is realized in terms of the weighted limit $lim^W F$ with the weight $W$ taken to be
where $K/k$ denotes the over category of $K$ over $k$ and $N(K/k)$ denotes its nerve.
This leads to the classical definition of homotopy limits in $\Simp\Set$-enriched categories due to
See for instance also
In some nice cases the weight $N(K/-)$ can be replaced by a simpler weight; an example is discussed at Bousfield-Kan map.
For instance in the case that $K = \{r \to t \leftarrow s\}$ is the pullback diagram we have
and $W(r \to t) : \{r\} \to \{r \to t \leftarrow s\}$ injects the vertex $r$ into $\{r \to t \leftarrow s\}$ and similarly for $W(s \to t)$.
This implies that for $F : K \to C$ a pullback diagram in the SimpSet-eriched category $C$, a $W$-weighted cone over $F$ with tip some object $c \in C$, i.e. a natural transformation
is
over $r$ a “morphism” from the tip $c$ to $F(r)$ (i.e. a vertex in the Hom-simplicial set $C(c,F(r))$);
similarly over $s$;
over $t$ three “morphisms” from $c$ to $F(t)$ together with 2-cells between them (i.e. a 2-horn in the Hom-simplicial set $C(c,F(t))$)
such that the two outer morphisms over $t$ are identified with the morphisms over $r$ and $s$, respectively, postcompoised with the morphisms $F(r \to t)$ and $F(s \to t)$, respectively.
So in total such a $W$-weighted cone looks like
as one would expect for a “homotopy cone”.
Details of this are discussed for instance in the book
To compare with the above discussion notice that
The functor
is discussed there in definition 14.7.8 on p. 269.
the $V$-enriched hom-category $[K,V]$ which on $V$-functors $S,T$ is the end $[K,V](S,T) = \int_{k \in K} V(S(k), T(k))$ appears as $hom^K(S,T)$ in definition 18.3.1 (see bottom of the page).
for $V$ set to SimpSet the above definition of homotopy limit appears in example 18.3.6 (2).
A standard reference is
In