In American English, a localized electrical failure is a power outage or electrical outage; one covering a neighborhood or wider area is a blackout. The preferred British term appears to be power cut.
If you are using your computer and it loses power from the mains, you would say the power went out or the power has gone out (go out by analogy to fire or light, meaning to be extinguished). When asked why you aren't typing by a passerby who hasn't noticed, you would similarly explain the power is out.
Absent a major disaster or malfeasance, power cuts to consumers due to shortages are uncommon, so there isn't common terminology surrounding such events. They may be described as rolling blackouts in the media or load shedding by engineers. Power shortage is not in common usage for failure events, only to describe the overall state of the electrical grid (e.g. if additional transmission lines are not built, the state will face a power shortage by 2035). When a shortage of current causes a temporary drop in voltage, the event is known as a brownout.