Briefly put, the (Co-)end is the universal wedge of a diagram.
Why is it called (co-)end? What is it an "end" of? By the universal property, it is in some sense the "universal end/finish" of some diagram but the same can be said about (co-)limits, so that doesn't seem too convincing to me.
I have no proof this is the real reason, but I came to believe the name "end" arose to give a nod at "limit": very old notes (before Mac Lane there weren't real textbooks around..) call "root" what we today call "limit".
For limits, the modern metaphor became a standard because it's more elegant and suggestive (I mean: root? Seriously?), and because "limits, when they exist, are unique" becomes a true statement in a category as well as in a Hausdorff space.
But! "Ends" have been a thing in general topology since Freudhental: https://en.wikipedia.org/wiki/End_(topology)
Given a topological space you can cook a functor ${\bf Top}_* \to\bf Set$ sending a space $X$ to its set of "ends", i.e. limits (in the categorical sense) of op-chains of $\pi_0$'s of complements of compact subsets of $X$ (whew, that was a nested definition).
Of course this begs a lot of questions (about the properties of that functor: coends have something to do with pi_0's, in the sense that the coend of $\hom_C$ is the $\pi_0$ of a certain arrow category of endomorphisms in $C$)! But most of all: how comes Freudhental defined something that is a functor (and not an easy one...), without having the word "functor" in mind?