Let a reductive group $G$ act on an affine variety $X$ (assume the base field $k$ is algebraically closed and of characteristic zero). Since the algebra of invariants $k[X]^G$ is finitely generated and does not contain nilpotents, the affine variety $X/\!/G:=Spec\, k[X]^G$ is defined. The morphism $\pi:X\to X/\!/G$ induced by the embedding $k[X]^G\to k[X]$ is constant on the orbits of $G$. Some additional, easy to check properties: $\pi$ is surjective, sends invariant closed sets in $X$ to closed sets in $X/\!/G$, and any fiber of $\pi$ is a union of orbits and contains a unique (Zariski-) closed orbit which is of minimum dimension among orbits in the fiber (this is rather elementary, you do not need Luna's slice etale for that). Note that $\pi^{-1}(\pi(x))=Cl(G\cdot x)$ for all $x\in X$.

In the projective case, results are different, as Allen pointed out in the example.

The results for non-algebraically closed fields, e.g., real algebraic actions, are slightly more complicated. I refer you to the excellent paper (I do not have time now, I will elaborate on that later if necessary)

MR1285780 (95f:14090) Reviewed Bremigan, Ralph J.(1-BLS) Quotients for algebraic group actions over non-algebraically closed fields. J. Reine Angew. Math. 453 (1994), 21–47. 14L30 (14D25) PDF Clipboard Journal Article Make Link

Let $k$ be a field of characteristic zero, not necessarily algebraically closed. The author studies the geometric invariant theory (GIT) quotients over a such a field. Let G be a reductive algebraic group acting on an affine variety V, both defined over the closure of k. The standard GIT quotient $V/\!/G$ exists. Let $(V_k,G_k)$ be the $k$-points of $(V,G)$. The author discusses the space $X=\pi(V_k)$ as a quotient object for the action of $G_k$ on $V_k$, where $\pi:V\to V//G$ is the GIT quotient map. If $k$ is local, there is a second possibility: the space $V_k/\!/G_k$ of closed $G_k$-orbits in $V_k$ is Hausdorff and supports a quotient map $p:V_k\to V_k/\!/G_k$. The main results are in $\S5$.

Let a reductive group $G$ act on an affine variety $X$ (assume the base field $k$ is algebraically closed and of characteristic zero). Since the algebra of invariants $k[X]^G$ is finitely generated and does not contain nilpotents, the affine variety $X/\!/G:=Spec\, k[X]^G$ is defined. The morphism $\pi:X\to X/\!/G$ induced by the embedding $k[X]^G\to k[X]$ is constant on the orbits of $G$. Some additional, easy to check properties: $\pi$ is surjective, sends invariant closed sets in $X$ to closed sets in $X/\!/G$, and any fiber of $\pi$ is a union of orbits and contains a unique (Zariski-) closed orbit which is of minimum dimension among orbits in the fiber (this is rather elementary, you do not need Luna's slice etale for that). Note that $\pi^{-1}(\pi(x))=\{y\in X| x\in cl(G\cdot y)\}$ if $G\cdot x$ is a closed orbit. Any orbit $\mathcal O$ is open in its closure, so $cl(\mathcal O)\setminus\mathcal O$ is a union of orbis of strictly lower dimension, see J. Humphreys book on Linear Algebraic Groups, Proposition in section 8.3.

In the projective case, results are different, as Allen pointed out in the example.

The results for non-algebraically closed fields, e.g., real algebraic actions, are slightly more complicated. I refer you to the excellent paper (I do not have time now, I will elaborate on that later if necessary)

MR1285780 (95f:14090) Reviewed Bremigan, Ralph J.(1-BLS) Quotients for algebraic group actions over non-algebraically closed fields. J. Reine Angew. Math. 453 (1994), 21–47. 14L30 (14D25) PDF Clipboard Journal Article Make Link

Let $k$ be a field of characteristic zero, not necessarily algebraically closed. The author studies the geometric invariant theory (GIT) quotients over a such a field. Let G be a reductive algebraic group acting on an affine variety V, both defined over the closure of k. The standard GIT quotient $V/\!/G$ exists. Let $(V_k,G_k)$ be the $k$-points of $(V,G)$. The author discusses the space $X=\pi(V_k)$ as a quotient object for the action of $G_k$ on $V_k$, where $\pi:V\to V//G$ is the GIT quotient map. If $k$ is local, there is a second possibility: the space $V_k/\!/G_k$ of closed $G_k$-orbits in $V_k$ is Hausdorff and supports a quotient map $p:V_k\to V_k/\!/G_k$. The main results are in $\S5$.