Mark the wall to indicate both the side edge and the top of the appliance. Add about 1/2 inch to the edge mark to allow for clearance around the appliance.
Pull the appliance out of position to provide work room. Tap on the wall that runs behind the appliance. Locate the positions of the studs -- the wood framing inside the wall -- behind the wall surface. When tapped on, the wall will sound and feel solid if a stud is present. If there isn't a stud directly behind the area where you wish to build your wall, you may build the new wall without anchoring it to the old wall. A full length wall is fine when secured only to the ceiling and floor, but a short wall may prove flimsy. Adding wood framing inside the wall solves this issue.
Mark the positions of the studs on either side of the new wall's location. Cut the wall covering off carefully. Typically, drywall requires only a utility knife but harder surfaces may require a reciprocating saw to penetrate. Score the wall material on the inside edge of the studs on either side of the area, then remove the material. Save the wall covering if you can reuse it, or discard it if you plan to install a new material when you're finished.
Measure the distance between the studs -- generally from 16 to 24 inches. Cut one board for every 8 to 12 inches of wall height you plan to build. Thus, a full height wall might require six or more boards, but a wall that's appliance-high might only need three or four.
Nail the boards between the studs on either side of the opening. Toenail the wood -- drive nails at an angle through the board into the studs -- with two or three nails on each end of each board. Make a light pencil mark on the wall, on either side of the opening, to indicate where the boards are; these will guide you later.
Recover the wall as desired. Nail drywall or paneling into place. Finish edges with trim, or drywall tape and drywall mud, to cover the seams.
Measure from the edge of a counter cabinet or similar item to the mark previously made, indicating the edge of the appliance plus clearance. Repeat at the bottom of the wall and near the top, as well, if you plan to build a full-height wall. Run a straightedge between all the marks to show the new wall's position.
Mark the floor with tape or a chalk line to indicate the wall position as well as the length of the wall, from the rear forward. Typically, the wall's length matches either the depth of the counter and cabinets on the other side of the appliance, or the depth of the appliance itself.
Cut two sections of 2-by-4-inch boards to the length of wall desired. These will comprise your sole plates, which sandwich the studs to form the wall. Substitute 1-by-2 boards for a thinner wall, or create a thicker wall by building two 2-by-4 walls.
Find the height of wall desired, whether appliance or countertop level (generally 36 inches), or floor to ceiling. Subtract the thickness of the plates, and the thickness of the countertop material as well if you plan to build a wall and top it with a short section of counter material. Take 1/2 inch extra off if the wall is ceiling high.
Each 2-by-4 plate is 1 1/2 inches thick; 1-by-2 plates are 3/4 inch thick. Countertops vary in thickness. So for a 2-by-4 wall 36 inches high, topped with a 1 1/2-inch-thick countertop: subtract 3 inches (plate thickness) and 1 1/2 inches (countertop thickness) from 36 inches to obtain 31 1/2 inches. This is your stud height.
Line the plates up, ends flush, and lay a square across both at the same time. Make a mark at 3 inches and every 8 inches thereafter until you reach the end. Mark 3 inches before the end. Cut one stud for each mark made, except at the beginning and end, where you'll insert a second stud.
Lay the plates, on the thin edge, parallel to each other. Attach two studs to one plate, then the other, at the beginning. Nail through the plate into the stud end on either side. Continue across the length, doubling the stud again at the end of the plates. Flip the boards over and attach the opposite plate. When complete, each stud is attached between the plates like a ladder.
Set the stud wall in place, lining it up with the guide marks. Hold a level against both the side and end to check for plumb -- it must be perfectly straight up and down. Adjust as needed. Insert shims under full length walls to compensate for the fact that the wall is slightly shorter than the ceiling.
Drive nails through the bottom plate into the floor every 4 to 6 inches. Zigzag the nails to create extra stability. Attach through the first studs into the wall bracing similarly, and through the top plate into the ceiling wherever a ceiling joist board is present.
Insulate the wall if desired. Finish with a wall covering on either side. Top it with a countertop, measuring the width of the wall plus a slight overhang away from the appliance. Attach it as instructed by the manufacturer. Push the appliance back into place to complete the job.