A commutative diagram is just a collection of nodes and directed edges between nodes (aka, a directed graph), but it's a directed graph along with the claim that any two paths in this graph which start at the same node and end at the same node are to be considered equivalent, in some sense.
In general, a directed (multi)graph along with an account of which of its paths are and are not to be considered equivalent to each other (where this equivalence relation satisfies some basic nice properties) is known as a "category". This concept comes up ubiquitously in math/abstract logic/etc. Commutative diagrams are useful for quickly visually reasoning about equivalences of paths in such contexts.
Note that Quiver isn't really a tool just for drawing "commutative diagrams". It's a tool for drawing any labeled system of nodes and edges/arrows between nodes, whether or not this is to be interpreted as a commutative diagram. (It also allows drawing arrows which start or end at other arrows, rather than at nodes).
It has various features to control and adjust these diagrams to be visually pleasing, by changing sizing/spacing/curviness/arrow style/etc of the elements within these diagrams. This is all much more convenient in its WYSIWYG interface than manually planning and coding these figures in LaTeX, as had previously been the standard way to create them for mathematical papers.
> This is all much more convenient in its WYSIWYG interface than manually planning and coding these figures in LaTeX, as had previously been the standard way to create them for mathematical papers.
While I'm all for such convenience tools, I thought HN would be the place to find sympathy for the idea behind tikz-cd and its predecessors like xypic, that it's easier to write code that can be easily reproduced and programmatically manipulated than it is to try to draw (by hand, or with something like xfig), which was previously to that the standard. I guess preferences are cyclical!
Sure, no knock on the tool; it's good for people to choose where they want to spend their technical understanding. Just funny to be old enough to have seen some cycles firsthand.
In general, a directed (multi)graph along with an account of which of its paths are and are not to be considered equivalent to each other (where this equivalence relation satisfies some basic nice properties) is known as a "category". This concept comes up ubiquitously in math/abstract logic/etc. Commutative diagrams are useful for quickly visually reasoning about equivalences of paths in such contexts.