nCatLab notation At this nCatLab page about the Reedy Model structure, a piece of notation is used which I do not understand (and appears not to be explained there). 
Suppose that we have a Reedy diagram, whose objects $X_r$ are indexed by the underlying Reedy category. Under the definition of the latching object for an object $X_r$ in this diagram, we have:

$L_r X = \text{colim}_{s \overset{+}{\to} r} X_s$.

I can have a guess, based on examples which I've seen in various places, what this could mean, but I would prefer a precise definition.
I think this means that the latching object for $X_r$ is defined the colimit of the diagram which we get by taking all objects and maps which go into $X$ which increase the degree. 
This is a bit of an educated guess, though, and even if this is correct I would like the precise meaning of the notation above. 
 A: Yes, your guess is basically right. The fundamental idea is that given two functors $X,Y \colon I \rightarrow \mathcal{C}$ from a small category $I$ we want to find recursively a transformation $\alpha \colon X \rightarrow Y$, then to do the recursive step we have to produce a morphism $\alpha_i \colon X_i \rightarrow Y_i$ which must be compatible with the other maps $\alpha_j$ for the values $j$ "coming before" $i$ (this coming before in the general category $I$ is specified by the Reedy structure). Then this $\alpha_i$ must fit in a diagram $\require{AMScd}$
\begin{CD}
L_iX @>{}>> X_i;\\
@VVV @VV{?}V \\
L_iY @>{}>> Y_i
\end{CD}
where on the left the latching object $L_iX$ is an object in $\mathcal{C}$ which attaches the $X_j$s for $"j<i"$ in a way that keeps tracks of the coherence conditions of the maps $X_j \rightarrow X_{k}$ coming from the functoriality of $X$. Then the maps $\alpha_j$ which you have by recursive assumption give rise to the map $L_iX \rightarrow L_iY$.
In analogy with CW complexes: If $X$ is a $CW$-complex then you can think of $I$ with its Reedy structure as an appropriate ordering of the cells which respects the dimensions, then $L_iX$ is the subcomplex you get by attaching the cells before the $i$-th one. And the matter of finding $\alpha$ is just a question of extending maps defined on the singular cells to a cellular map.
Since you know about model structures I think you can imagine why you would want to find such transfomations $\alpha$: the modus operandi of model categories revolves around producing lifts fitting in appropriate square diagrams.
As extra: here some notes by Dugger in which one of the sections he discussed about the Reedy model structure, see page 56.
https://pages.uoregon.edu/ddugger/hocolim.pdf
Surely it is not the most complete discussion about this argument, but I think that it is really accessible and intuitive.
