It is well-known that the category of schemes is not cocomplete (i.e., colimits does not necessarily always exist). So I am looking for a counterexample whenever we have an algebraic group $G$ acting "nicely" on a scheme $X$, such that the quotient $X/G$ (although it's locally ringed space) is not a scheme. Because I wasn't able to find out a counterexample could you provide one. What's the main reason behind this problem?
P.S. I guess if it acts sufficiently "nice"on $X$ then the resulting space will be a scheme eventually, therefore, otherwise the question can be rephrased as: what's the minimum prerequisite for an action not being "nice" enough to give at $X/G$ a scheme structure.
You should be able to pick any pair $G,X$ with an action of $G$ on $X$ which is not free. Easy examples would be $\Bbb G_m$ acting on $\Bbb A^n$ by dilation. There are other examples ($G$ not {finite,locally free} or possibly if $X$ is badly behaved enough) but this failure of the action to be free is probably the most accessible example.