Is there any example of a Category without generators? I was looking of an example of a category (a non trivial example) without generators (seperators). Is there any nice example? Or every category that we are using has generators?
 A: It depends on what you mean by "generators." There is a notion of a collection $S$ of objects being a "family of generators" of a category $C$, which means that the functors $\text{Hom}(s, -) : C \to \text{Set}, s \in S$ are jointly faithful. Any category has a (possibly large) family of generators given by taking every object, so the interesting question is when a category has a small (set-sized) family of generators. 
An example of a category, even an abelian category, without this property is the category of ordinal-graded vector spaces (that is, collections $V_{\alpha}$ of vector spaces, one for each ordinal) with bounded support (that is, for all objects $V_{\bullet}$ there is some $\alpha$ such that $V_{\beta} = 0$ for all $\beta > \alpha$). Any set-sized collection of objects $S$ collectively has bounded support (namely the supremum of the supports of the objects in $S$) and so cannot detect $V_{\bullet}$ with support above that bound. 
The property of admitting a small family of generators implies and is closely related to the property that a category is concretizable. 
A: This is perhaps not the most natural example, but the category of groups doesn't have a cogenerating set (se here for why), so $\mathrm{Grp}^{\mathrm{op}}$ can't have a generating set.
