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 $J$?)

The cone is indexed by all objects of $J$.
Before dealing with limits in general, you should understand products and their universal property. Of course all factors are involved.
