Parameterizing a surface means placing coordinates for points all along the surface directly on the surface via a function Φ:D⊂(ℝ^2)⟶(ℝ^3).  Here, the domain points (u,v) become points on the surface via the map Φ.  Note that this surface cannot be written as z=f(x,y), due to the fold in the image.  However, it can be written as Φ(u,v)=(u,((3((v^3)-v))/4), ((2v)/5)), for u∈[-2,2],  and v∈[-1.5,1.5].  Follow the dot in the image as you move the dot in the domain.

surface = ParametricPlot3D[{u,.75(v^3-v),.4v},{u,-2,2},{v,-1.5,1.5},Mesh->None];
plane = Graphics[{LightGray,Rectangle[{-2,-1.5},{2,1.5}]},PlotRange->{{-2.5,2.5},{-2,2}},Axes->True,AxesOrigin->{0,0}];
rpoint[u_,v_]:= Graphics[{Red,PointSize[Large],Point[{u,v}]}];
rpoint3d[u_,v_]:=Graphics3D[{Red,PointSize[Large],Point[{u,.75(v^3-v),.4v}]}];
Manipulate[{Show[plane,rpoint[u,v]],Show[surface,rpoint3d[u,v],Boxed->False,AxesOrigin->{0,0,0},ImageSize->400]},{u,-2,2},{v,-1.5,1.5}]

Here we parameterize the unit sphere using two angular coordinates, just like latitude and longitude on our globe.  However,the extent of our two variables is  u∈[0,2π], and v∈[0,π].  Here Φ:D⊂(ℝ^2)⟶(ℝ^3) is given by Φ(u,v)=(cos u sin v,sin u sin v,cos u).  Again, follow the dot in the image as you move the dot in the domain.

surface2= ParametricPlot3D[{Cos[u]*Sin[v],Sin[u]*Sin[v],Cos[v]},{u,0,2*Pi},{v,0,Pi},Mesh->None,PlotStyle->Opacity[.5]];
plane2 = Graphics[{LightGray,Rectangle[{0,0},{2*Pi,Pi}]},PlotRange->{{-.5,7},{-.5,4}},Axes->True,AxesOrigin->{0,0}];
rpoint2[u_,v_]:= Graphics[{Red,PointSize[Large],Point[{u,v}]}];
rpoint3d2[u_,v_]:=Graphics3D[{Red,PointSize[Large],Point[{Cos[u]*Sin[v],Sin[u]*Sin[v],Cos[v]}]}];
Manipulate[{Show[plane2,rpoint2[u,v]],Show[surface2,rpoint3d2[u,v],Boxed->False,AxesOrigin->{0,0,0},ImageSize->400]},{u,0,2*Pi},{v,0,Pi}]

And here we parameterize the torus, again using two angular coordinates, The extent of our two variables is  u∈[0,2π], and v∈[0,2π], and the map is Φ:D=[0,2π]×[0,2π]⊂(ℝ^2)⟶(ℝ^3) is given by Φ(u,v)=((2+cos u) cos v,(2+cos u )sin v,sin u).  Once again, follow the dot in the image as you move the dot in the domain.

surface3= ParametricPlot3D[{(2+Cos[u])*Cos[v],(2+Cos[u])*Sin[v],Sin[u]},{u,0,2*Pi},{v,0,2*Pi},Mesh->None,PlotStyle->Opacity[.5]];
plane3 = Graphics[{LightGray,Rectangle[{0,0},{2*Pi,2*Pi}]},PlotRange->{{-.5,7},{-.5,7}},Axes->True,AxesOrigin->{0,0}];
rpoint3[u_,v_]:= Graphics[{Red,PointSize[Large],Point[{u,v}]}];
rpoint3d3[u_,v_]:=Graphics3D[{Red,PointSize[Large],Point[{(2+Cos[u])*Cos[v],(2+Cos[u])*Sin[v],Sin[u]}]}];
Manipulate[{Show[plane3,rpoint3[u,v]],Show[surface3,rpoint3d3[u,v],Boxed->False,AxesOrigin->{0,0,0},ImageSize->400]},{u,0,2*Pi},{v,0,2*Pi}]

And finally, choose a point p=((Subscript[x, 0]),(Subscript[y, 0]),(Subscript[z, 0]))∈ (ℝ^3)and two non-zero, noncolinear vectors α =(([(Subscript[α, 1]),(Subscript[α, 2]),(Subscript[α, 3])])^T)and β=(([(Subscript[β, 1]),(Subscript[β, 2]),(Subscript[β, 3])])^T), based at p.  Then,defining γ as the vector based at the origin with head at (Subscript[x, 0]), we can parameterize the plane spanned by α and β by 
Φ:D=(ℝ^2)⟶(ℝ^3), Φ(u,v)=γ+ uα + vβ=((Subscript[x, 0])+u (Subscript[α, 1])+v (Subscript[β, 1]),(Subscript[y, 0])+u (Subscript[α, 2])+v (Subscript[β, 2]),(Subscript[z, 0])+u (Subscript[α, 3])+v (Subscript[β, 3])).  
Below is a piece of this plane.  Below, we choose p=(2,3,1),α =(([1,1,1])^T) and β=(([-1,2,0])^T), but feel free to change these to explore....

p = {2,3,1};
alpha = {1,1,1};
beta = {-1,2,0};
surface4= ParametricPlot3D[{2+u-v,3+u+2v,1+u},{u,-10,10},{v,-10,10},Mesh->None,PlotStyle->Opacity[.5]];
plane4 = Graphics[{LightGray,Rectangle[{-10,-10},{10,10}]},PlotRange->{{-11,11},{-11,11}},Axes->True,AxesOrigin->{0,0}];
rpoint4[u_,v_]:= Graphics[{Red,PointSize[Large],Point[{u,v}]}];
rpoint3d4[u_,v_]:=Graphics3D[{Red,PointSize[Large],Point[{2+u-v,3+u+2v,1+u}]}];
Manipulate[{Show[plane4,rpoint4[u,v]],Show[surface4,rpoint3d4[u,v],Boxed->False,AxesOrigin->{0,0,0},ImageSize->400]},{u,-10,10},{v,-10,10}]