Given a partial order $\langle X, \leq \rangle$ and a subset $I \subseteq X$ it is common to consider $I$ as the generators of the set $\{ x \in X: i \leq x \textrm{ for some }i \in I \}$ (i.e., the up-set generated by the elements of $I$).
On the other hand, partial orders can be seen as a very particular simple type of categories: those ones where for every two objects there is at most one arrow among them.
I wonder which are the categorical generalizations of the previous "generator" notion. In other words, is there some well known notion(s) for arbitrary categories which plays the role of "generating"?
You can consider pretty much the same thing; e.g. given a set $X$ of arrows, you can consider the set of arrows $\{ x \circ f \mid x \in X \} $, or the reverse, or a two-sided thing.
A particularly notable special case is that of a sieve, where we consider the set generated as above, but all of the arrows of $X$ have codomain to be a single object of the category.
The notion of a Grothendieck topology on a category involves sieves: a topology is a way of specifying which sieves on an object count as a "cover" of that object. If you have a generating set for the sieve, you can imagine each of the generators as specifying one set in the covering. (and all of the other arrows in the sieve are just throwing in all of the refinements of the covering, in a similar way that in ordinary topology, one usually talks about a topology, rather than a basis for the topology)
There is a standard term "generator" in category theory, but it's unrelated to the notion you describe: a generator is a set of objects (often a single object) with the property that if $f,g : A \to B$ are two different arrows, then there is an arrow $h:X \to A$ where $X$ is in the generating set with the property that $f \circ h \neq g \circ h$.
One way to interpret how this "generates" a category is that you can imagine replacing each object $A$ of the category with the set of all arrows $X \to A$ where $X$ is in the generating set. We can also interpret morphisms as ordinary functions on these sets (the function you get by composition).
This construction gives a functor, and the condition to be a generating set says that it is a faithful functor. So, the generating set can be thought of as generating the category in the sense that it produces "enough elements" to be able to properly distinguish the structure of the category.