Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$? 
Let $F : J → C$ be a diagram of type $J$ in a category $C$. A cone to
  $F$ is an object $N$ of $C$ together with a family $ψ_X : N → F(X)$ of
  morphisms indexed by the objects $X$ of $J$, such that for every
  morphism $f : X → Y$ in $J$, we have $F(f) \centerdot ψ_X = ψ_Y$. http://en.wikipedia.org/wiki/Limit_(category_theory)

So, is $X$ here referring to a single object of $J$ or several objects of $J$ grouped together? (In terms of set, is $X$ an element of $J$ or a subset of $J$?)

There is a collection of $\psi_X$. One $\psi$ for each object in $J$. So if $J$ has 3 objects $a,b,c$ you have $\psi_a, \psi_b,\psi_c$. But, in general, you cannot choose the $\psi$ as you wish.  The various $\psi$ have to respect some constraints among themselves: if there is a $f:X \to Y$ then, even if you can freely choose $\psi_X$,  $\psi_Y$ is automatically determined by $\psi_Y = F(f) \centerdot \psi_X$.
So, to answer explicitly to the question in your title: each $X$ is a single object in $J$. The collection of $X$ is the collection of all objects in $J$
