Here a new receptacle is added to a light fixture circuit. The connections at the light fixture are spliced and extended to the new receptacle. With this arrangement the new receptacle will turn off and on with the light fixture.
Here 3-wire cable serves a split receptacle. The bottom half of the receptacle is controlled by a switch and the top half is always-hot.
Here a switch has been added to control an existing receptacle. The switch takes the hot from the middle receptacles. With this arrangement, all the other receptacles are wired directly to the hot source and the switch controls the middle outlet.
The neutral wire in this circuit is taken from the receptacle and spliced to the white on the new cable, and to a pigtail that runs back to the receptacle. The hot is spliced to the black cable wire and a pigtail back to hot on the receptacle.