What actually fails when the filter stops
A filter does three jobs at once, and a stoppage removes all three on different clocks. Water movement ends immediately, and with it the surface exchange that keeps oxygen entering the water. The bacteria living in the filter media — the colony that converts fish waste into far less harmful substances — depend on that flow for their own food and oxygen, and a colony starved of flow begins to decline within hours. Meanwhile the fish keep producing waste exactly as before. Nothing dramatic happens in the first minutes; the danger builds quietly and at different speeds in different tanks.
The hours depend on stocking and warmth
A small number of hardy fish in a large, planted tank may be untroubled by a stoppage lasting much of a day, because the water volume dilutes waste and the plants contribute oxygen in the light. A crowded community tank, a tank of large messy fish, or any tank in warm weather sits at the other extreme: warm water holds less oxygen while warm fish need more of it, and trouble can arrive within a few hours. Judge your own tank honestly rather than borrowing a number from someone else's — the fuller and warmer it is, the shorter your margin.
Keeping oxygen up without power
While the filter is off, your main job is oxygen. Stop feeding at once: digestion uses oxygen, and uneaten food makes everything worse. Every so often, stir the surface or pour a jug of tank water back in from a height to ripple and splash it — crude, but it genuinely moves oxygen into the water. A battery air pump, if you have one, does the job better and is worth its place in the cupboard. Keep the lid closed in cold weather to hold heat in, since a chilled tank stresses fish almost as much as a stuffy one; in a heatwave, do the opposite and leave the lid ajar with the lights off.
Protect the filter media while you wait
If the stoppage will last more than a short while, rescue the bacteria even if you cannot run the filter. Take the media out and keep it submerged in a bucket of the tank's own water — media left in a drained filter dries out, and a dried colony is a dead one, which means the tank restarts almost as if uncycled. Never wash the media in tap water to help during the outage; chlorine damages the very colony you are trying to save. A sponge soaked and resting in the bucket of tank water keeps its bacteria wet and fed for far longer than one sitting dry in a silent filter box.
Restarting and watching the days after
When power returns, check the filter actually restarts — impellers often need a nudge after sitting still — and expect the water to look hazy as flow resumes. Feed lightly for the first day or two, because the colony may have thinned, and test the water if you have a kit, watching for ammonia in the days after a long stop rather than assuming the danger ended with the outage. One careful water change does more good than any bottled quick fix. For a stoppage in a full, warm tank, put a follow-up question to KinsLe's Kimmy; she answers using the KinsLe Answers library.