Make the hole in the wall
Determine where the studs are in your wall by driving a nail through the drywall. Do this in the center of where the niche is going, so the nail holes you create will be removed. The centers of the studs are probably 16 inches apart, making the space between them 14 1/2 inches. When you drive a nail through the drywall in the center of where the niche is going, and don't encounter the resistance of a stud, put a drywall saw through the hole and cut horizontally in each direction until you hit a stud. Be careful not to cut through the other side of the wall.
Cut vertically with the drywall saw along the edge of each stud, to the top and bottom points of where the niche is going.
Cut horizontally at these points and remove the piece of drywall.
Build the niche
Cut two pieces of two-by-four to 14 1/2 inches long.
Install the bottom two-by-four by positioning it horizontally in the hole between the two studs, then pressing it down so that its top surface is even with the bottom of the hole. Secure this two-by-four by driving screws in diagonally through it and into the stud on each side.
Install the top two-by-four by doing the same thing as the previous step, at the top of the hole.
Cut pieces of drywall to fit the four inside faces of the hole you have created, and attach them to these faces using a drill and drywall screws.
Apply a thin coat of spackle to all the joints and cover them with drywall tape. Cover the tape with another coat of spackle. Leave it to dry for at least a day.
Sand the spackle to remove most of the unevenness, but don't sand so much that you expose the tape. Apply another coat of spackle, making it as smooth as possible, and allow it to dry.
Sand the second coat of spackle smooth.
Prime all the new surfaces.
Paint the niche and the surrounding wall, either the same color, or different colors to accent the niche.