A category consists of a collection of objects together with morphisms between these objects. Thus, naively, we may think of objects as the ‘elements’ of a category.
More generally, in higher category theory the objects of an $(n,r)$-category are the $0$-dimensional cells of that structure, the $0$-morphisms.
If a set is regarded as a discrete category (with no nontrivial morphisms) then the objects of that category are precisely the elements of the set.
In the fundamental groupoid $\Pi_1(X)$ of a topological space $X$, the objects are the points of $X$.
In the category Set, the objects are sets; in Vect the objects are vector spaces; in Top the objects are topological spaces, etc.
If a simplicial set that is a Kan complex is regarded as an $\infty$-groupoid, then its vertices are the objects of that $\infty$-groupoid.
Similarly if a simplicial set that is a quasi-category is regarded as an $(\infty,1)$-category, then its vertices are the objects of that $(\infty,1)$-category.
If a globular set is equipped with the structure of a strict ∞-category, then its $0$-cells are the objects of that ∞-category.