Find the nearest vertical stud in the area you want to build your pony wall. Because the pony wall needs to attach to a wall stud, you are limited somewhat in your placement by the location of the nearest vertical stud. Determine the center of the stud with the stud finder and mark it with a pencil and level.
Cut the framing pieces for your pony wall on the miter saw. You need horizontal pieces for the bottom and top spanning the distance you want your pony wall to be. You also need vertical pieces for the sides as well as for the studs in between. These should be 16 inches apart, so cut as many as you need according to the dimensions of your pony wall.
Mount the bottom piece on top of the tile floor. Use self-tapping screws and the drill to go down through the tile into the subfloor beneath. Set the screws every few inches across the entire piece. Mount your wall piece. Use 4-inch wood screws to go through the drywall into the wall stud. Use the level to guide the plumb of this piece.
Set your stud pieces along with the outer face piece that defines the outer perimeter. Screw the pieces into the base with the 3-inch wood screws. Angle the screws along the base of the vertical pieces to sink the screws into the base. Use the level to guide their plumb.
Set the top piece. Use the 3-inch wood screws to screw directly down into the vertical supports. Keep the overall structure plumb with the level. Cut down drywall pieces to fit with a utility knife and screw them into the framing of the pony wall with the drywall screws. Cover the pony wall as you see fit, with whatever finish material you want.