You want pigtails, you got pigtails. These pigtails are going to be attached to screw terminals on a temporary exposed bulb light fixture. So, I've pre-bent the end of each pigtail for the screw terminal.
The black wires show a pigtail that's done. The incoming hot wire is twisted together with the outgoing hot wire and a hot pigtail.
The white wires show how they're twisted together before installing the wire nut. The insulation ins stripped to the proper length on each wire. The three wires are then twisted tightly together using a lineman's pliers. If the ends of the wires don't line up post-twisting, then cut off the excess so they do. In this one I got lucky and they all lined up.
Note: Always twist the wires together - you'll get a better connection that won't come apart under slight wire movement or expansion/contraction due to changes in temperature. (You do know that flowing electricity makes wires get hot, right? ;-)
You'd use this same setup if you were wiring an outlet in the middle of a run (except you'd have a pigtail for the ground wire as well). Generally you want to do that so that a failure in the device/fixture doesn't cause everything downstream to stop working also...
For the previous post, I didn't need the pigtail to connect a device with the incoming and outgoing wires. Rather, the pigtail was just to extend the wire length. In that case there was only one wire twisted together with the pigtail.
Disclaimer (I work with lots of lawyers and they won't be happy unless I add a disclaimer): I'm not an electrician. I don't play one on TV. I didn't sleep at a Holiday Inn Express last night. Use at your own risk. Your mileage may vary. Always turn off the breaker before working on a circuit.