Two things would happen if the sun somehow stopped producing light and heat all of a sudden. First, the earth would begin to steadily cool down without the heat of the sun warming it. Within a few days the surface of the earth would be below freezing. Large bodies of water like the oceans cool down much slower than land, so the oceans would remain warmer for a time, but the surface of the ocean would eventually freeze after several months, but the water underneath the surface probably won’t freeze for a very long time. At this point the surface of the earth would be too cold for most plants and animals to survive and there would be no available liquid water to drink. After the oceans freeze, the earth will continue to cool for a thousand years or so until it reaches a temperature of -400 degrees Fahrenheit, at which point the atmosphere would be unstable and will collapse.

The second thing that would happen is that photosynthesis would immediately stop Plants will no longer be able to grow without the energy of the sun. Most plants would die within a few weeks,? but some trees can slow down their metabolism in winter and could survive for quite some time without sunlight. Eventually, after several decades of no sunlight, even large tress would begin to die without photosynthesis. Since plants are a major food source in the earth’s ecosystem, most of the animals would die as soon as their food source is gone. Scavengers would be the last to survive since they survive on dead and decaying organisms; howevermost life would perish within the first year.

In the deep oceans, near some geothermal vents, and insulated by the frozen layer of water at the surface, some chemosynthetic bacteria and archaea who never see much of the sun anyway and rely on geothermal energy could continue to survive for many 1000s of years without any problem, so life-itself could last quite a while, but most of the organisms that live on the surface of the earth would not survive past one year.

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Depends on what you mean by "out". If you were to turn off nuclear fusion in the sun, it would probably be about ten thousand years before anybody on Earth would notice. This is because the sun is very big, so it would take time for it to cool off, and because the reason why the sun is bright is because it is hot. That heat comes from nuclear fusion, but the light that we see is not from the core where the nuclear reactions take place, but from the surface.

Even if the sun were to go out, however, there are hot springs and other hydrothermal vents created by volcanic activity that still provide enough heat and chemical energy for some forms of life to exist. Removing the sun would not destroy these vents, even though the rest of the Earth would freeze. These vents will probably continue to exist for billions of years, as they have existed for billions of years already.