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)
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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 $\S5$.