The first part of the arch-building process is the planning stage. Decide where you want to place the archway, as well as its shape. The shape of the arch is determined by personal preference, but also by the width of the doorway. For instance, if the doorway is narrow, the arch should have a steeper arc along the bottom of the frame.
Take the measurements of the door frame. The arch will be installed directly against the header board across the top of the doorway and the jack studs on the sides. On a finished wall, remove the drywall or molding so you can have direct access to the boards. Create the arch faces by drawing an arch onto a sheet of 1/2-inch plywood, then cut the board. Use that as a template to create a second, identical arch face. The arch faces will be nailed to each side of a U-shaped frame that's constructed from 2-by-3 boards. The top board is the same width as the doorway, and the two side boards are 2 inches shorter than the side edges of the arch faces. The arch faces are nailed to the frame, one on each side.
The assembled arch framing is lifted into position against the door frame, where it's nailed to the frame. Cut blocks of 1-by-2 boards to fit inside the space between the two arch faces along the bottom of the framing. These boards are attached to the arch facing with drywall screws, and provide a place for you to attach drywall along the bottom of the framing. A strip of 1/4-inch drywall is dampened with water so it can be bent without cracking or ripping when it's installed along the bottom of the arch. Two drywall screws are driven through the drywall and into the 1-by-2 blocks that were installed along the bottom of the arch.
After the arch framing has been installed, and drywall attached along the bottom edge, flexible corner beads are stapled to the bottom edges of the arch, where the plywood of the arch faces meets the 1/4-inch drywall strip. The flexible beads smooth the edges of the frame. Joint compound is then applied to the plywood with a putty knife, and smoothed as much as possible with the knife. The joint compound is sanded smooth after it's dried, then two more applications are placed on the framing, each one being sanded smooth. The arch frame is then covered with a coat of primer, then with two coats of paint after the primer has dried. If you're painting the surrounding walls as part of a larger project, paint them at the same time you paint the arch to provide a smooth, uniform application of paint.