(Affine GIT quotient)
Let \(G\) be a reductive group over an algebraically closed field \(k\) acting on an affine variety \(X = \Spec A\). The ring of invariants \(A^G\) is finitely generated, and the GIT quotient is \(X /\!\!/ G = \Spec A^G\), with the morphism \(\pi \colon X \to X/\!\!/G\) induced by \(A^G \subseteq A\).
\(\pi\) is surjective and constant on orbits, and \(\pi(x) = \pi(y)\) exactly when the orbit closures \(\overline{Gx}\) and \(\overline{Gy}\) meet. Each fibre contains a unique closed orbit, so the closed points of \(X/\!\!/G\) correspond to closed \(G\)-orbits in \(X\).
(Projective GIT quotient)
If \(G\) acts on a projective variety \(X\) with a \(G\)-linearized ample line bundle \(L\), a point \(x\) is semistable if some invariant section \(s \in H^0(X, L^m)^G\) has \(s(x) \neq 0\). The GIT quotient is \(X /\!\!/_L G = \Proj \bigoplus_{m \geq 0} H^0(X, L^m)^G\), a projective variety receiving a morphism from the open set \(X^{ss}\) of semistable points.
Let \(\GG_m\) act on \(\AA^2\). With weights \((1,1)\), \(k[x,y]^{\GG_m} = k\) and \(\AA^2/\!\!/\GG_m\) is a point, because every orbit closure contains the origin. With weights \((1,-1)\), \(k[x,y]^{\GG_m} = k[xy]\) and \(\AA^2/\!\!/\GG_m = \AA^1\); the fibre over \(0\) is the union of the two axes and contains three orbits, of which only the origin is closed. With weights \((1,1)\) and the linearization \(\OO\) twisted by the identity character, the semistable locus is \(\AA^2 \setminus \{0\}\) and the projective quotient is \(\PP^1\).