The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes and "Abstract and Concrete Categories The Joy of Cats" by Adamek, Herrlich, and Strecker.
$\color{Green}{Background:}$
[From: Adamek, Herrlich, and Strecker]
$\textbf{Definition 1:}$
A $\textbf{source}$ is a pair $(A,(f_i)_{i\in I})$ consisting of an object $A$ and a family of morphisms $f_i:A\to A_i$ with domain $A,$ indexed by some class $I.$ $A$ is called the $\textbf{domain of the source}$ and the family $(A_i)_{i\in I}$ is called the $\textbf{codomain of the source}.$
$\textbf{Definition 2:}$
A $\textbf{sink}$ is a pair $((f_i)_{i\in I}.A)$ [sometimes denoted by $(f_i,A)_I$ or $(A_i\xrightarrow{f_i} A)_I$] consisting of an object $A$ (the $\textbf{codomain}$ of the sink) and a family of morphisms $f_i:A_i\to A$ indexed by some class $I.$ The family $(A_i)_{i\in I}$ is called the $\textbf{domain}$ of the sink. Composition of the sink is defined in the (obvious) way dual to that of composition of sources.
[From: Arbib and Manes]
$\textbf{Definition 3:}$
$\quad$A $\textbf{cone}$ for a diagram $D$ is a family $X\rightarrow D_i$ of morphisms from a single object $X$ such that $X\rightarrow D_i\rightarrow D_j=X\rightarrow D_j$ for every $D_i\rightarrow D_j$ in $D$. A morphism from a cone $(X\rightarrow D_i)$ to a cone $(X'\rightarrow D_i)$ is a $\textbf{K}$-morphism $X\rightarrow X'$ such that $X\rightarrow X'\rightarrow D_i=X\rightarrow D_i$ for all $i$. The cones for $D$ then form a category, and a $\textbf{limit for the diagram}$ $D$ is a $\textit{terminal }$ object in this category, i.e., $(X\rightarrow D_i)$ is such that for all cones $(X'\rightarrow D_i)$ on $D$ there is a $\textit{unique}$ morphism $X'\rightarrow X$ such that $X'\rightarrow X\rightarrow D_i=X'\rightarrow D_i.$ We say that a limit $(X\rightarrow D_i)$ for $D$ has the $\textbf{universal property}$ with respect to cones $(X'\rightarrow D_i)$.
$\color{Red}{Questions:}$
I don't understand the point of the definitions of source and sink defined above. I know that the notion of cone and cocone make use, respectively of of the family of morphisms defined in the definition for the notion of source and sink. So for both source and sink, it involves an object and a family of morphisms. With each map in the family of morphism, either the object is the domain or is the codomain of each map. But other than that, I don't know why in category theory, there needs to be notions for "source" and "sink". Thank you in advance.
Best Answer
The important thing about a source is that it is a family of morphisms which all start at the same object. Similarly a sink is a family of morphisms which all go to the same object. I don't think there is anything more to understand here.
But it might be worth mentioning, why one would find these things so important to give them a dedicated name. One usecase for these notions is for cones over and cocones under a diagram: those are sources respectively sinks, which satisfy some compatibility conditions. (Note that these compatibility conditions are so important, that category theorists introduced even more terminology: cones and cocones)
Another usecase of say sinks, would be to describe coverings. By this I mean a jointly surjective/epimorphic family of maps $(f_i:U_i\rightarrow X)_i$. Things like that appear as soon as you do anything related to topology or geometry. Again, the additional conditions arising in these contexts will lead to even more terminology (coverings / sieves).