Would I mind? In a certain sense yes, because you are supposed to strive for explanations yourself. For instance look up the descriptions of the functions used.

But ok, here we are

p0 = {0, 0, 0};  (* Origin  *)
p1 = {1, -1, 2}; (* Position of Head of Arrow  *)
p2 = {1, 0, 0};  (* Three points defining your plane *)
p3 = {1, 1, 1};
p4 = {0, 0, 1};
gr1 = Graphics3D[{Thick, Arrow[{p0, p1}], PointSize[.05], 
    Point /@ {p2, p3, p4}
    }, Axes -> True, 
   AxesLabel -> {"X", "Y", 
     "Z"}]; (* Graphics. my version Mma 7 doesn't know InfinitePlane, so I changed it to Point  \
*)
v = p1 - p0;   (* Head of vector. equals p1, therefore v is not \
necessary*)
n1 = p2 - p3; (*basevectors of plane, better n1=p3-p2*)
n2 = p3 - p4;(* better p4-p2 *)
n = Cross[n1, n2]; (* normal to plane  *)
pl = p0; (* not necessary, , pl equals p0 = {0,0,0}  *)
pp = p2;  (* not necessary, pp equals p2  *)
equs = Thread[pl + lambda v == pp + mu n1 + nu n2];
(* pl + lambda v = {0,0,0} + lambda p1 is the equation of a line thru \
{0,0,0 } in direction of your vector  *)
(* if this line reaches the plane it must fulfill the equation of the \
plane: lambda v == plane =p2+....  . Thread makes three equations out \
of one *)
sol = Solve[equs, {lambda, mu, nu}][[1]];
(* solve it vor lamda , mu, nu  *)
pb = p0 + lambda v /. sol; 
(* calculate the point where the vector meets the plane. could be \
done as well by plane/.sol  *)
vern = n/Norm[n];  (* normalized normal to plane *)
prjn = (v.vern) vern;  (*Projection of vector to normal *)
prjP = v -  prjn; (* vec orthogonal to normal,  lying in the plane: prjP.n ->0, prjP + \
prjn gives v = vector, so you have vector decomposed in a component \
lying in the plane and one component parallel to the normal of the \
plane *)
gr2 = Graphics3D[{Thick, Dashed, Red, Arrow[{pb, prjP + pb}]}];
Show[gr1, gr2]

No idea. Depends on your session. Seems somewhere you have redefined sol. Try

Clear[sol]
sol = Solve[
   Join[Thread[
     Cross[pp - plane, normal] == {0, 0, 0}], {normal.(pp - plane) == 
      normal.normal tau}], {u, v, tau}] // Flatten

Fine that I could help you and things became clearer.

should look like the picture I posted above

You should play around with the u and v in the Plot3D-command, or turn the plot around with the mouse until it fits, or try this (play around with the coordinates of ViewPoint and see what happens)

Show[
 ContourPlot3D[
  normal.{x, y, z} == normal.plane, {x, -2, 2}, {y, -2, 2}, {z, -2, 
   2}, ContourStyle -> Opacity[.5], ViewPoint -> {1.3, -5.4, 2}],
 Graphics3D[{Red, PointSize[.04], Point[px], Point[xx], Point[pp], 
   Arrow[{px, xx}], Arrow[{pp, xx}], Blue, PointSize[.06], 
   Point[{0, 0, 0}], Thickness[.01], Arrow[{{0, 0, 0}, pp}]}], 
 PlotRange -> All]

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Hey, thank you for the very detailed answer. Everything is very clear and I now understand how to get to projection onto the plane.

But there is one thing though, in the exercise my 3D plane should look like the picture I posted above. That's what I need help with now. Thanks in advance :)