You're confusing two subtly different things.
Residential power in North America is usually delivered as "split phase", +120V, neutral, and -120V. (180 degrees out of phase: https://www.youtube.com/watch?v=vOh2OSJ44eE) The voltage across the two hot conductors is 240V, which is used for big stuff. You get it by tapping a transformer in the middle of a coil: https://en.wikipedia.org/wiki/Split-phase_electric_power
Three phase power has three hot conductors, 120 degrees out of phase to each other. In the US it can be delivered at a bunch of different voltages, depending on what transformer the customer installs, but generally I worked with 277/480. (277V hot to neutral, 480V hot to hot. Generally all commercial lighting runs on 277V.)
And to be specific, three phase is primarily useful because motors that use it are self-starting, you don't need to capacitor-and-switch arrangement single phase motors use, which frequently break. Less important now that most big motors use VFDs. Also, supposedly due to some frequency voodoo you don't need to use as much cable when running three phrase transmission cable.