The light of the sun
We wouldn't even notice the second she'd fade away. I can hear you thinking 'How is that possible?'. Well, light from the sun takes eight minutes and twenty seconds to reach the earth. For a little more than eight minutes we would just live our lives as we do now, but after those eight minutes panic and confusion breaks out because the sun is no longer visible.
Also the earth would lose the gravity of the sun and the earth will shoot out of the orbent in a straight line from the place where it was at that moment around the sun. From the earth we could still see a number of planets, like Jupiter. Jupiter will be able to reflect the sunlight for about 30 minutes after the sun has gone. Depending on the position in which Jupiter was in, it might reflect an extra 30 minutes before we would see the light dim until there is nothing left of it.
We no longer have moon and sunlight. We now only have space as a source of light, which fortunately gives off some light so that we can see just a little bit. With some luck our eyes would adjust, but the world will be quite dark and look like night.
As you probably learned in biology class, plants and trees need carbon dioxide, water and sunlight to produce oxygen and sugar, this is called photosynthesis. When there is no more sunlight there will be no photosynthesis. This means that most plants die within days and weeks. However, very large plants or trees have so much sugar in them that they can theoretically live for years without sunlight. The only problem for the larger plants and trees is that the soil will also cool down significantly, which make them freeze to death before they reach the end of their sugar supply.
And what about the oxygen? Since the plants no longer produce oxygen you would think that people will suffocate, but fortunately this is not the case. The amount of people on earth today together breathe in about 6,000,000,000,000,000 kg of oxygen per year, but our atmosphere contains 1,000,000,000,000,000,000 kg of oxygen. What a number, I'm not going to try to pronounce it!
This is enough to supply the 7 billion people and all animal species or life forms that consume oxygen with oxygen for thousands of years.
The sun is one of the Earth's largest heat sources. The Earth's surface area is currently on average about 15 degrees (57 degrees F). It will get colder much slower over time, but in the first few years after the sun is no longer a heat source, the average temperature will drop very quickly. After a week the temperature will be around freezing point.
After a year the temperature will have dropped to -73 degrees Celsius (-100 degrees F). At this point, many people will have died from the extreme cold. But there is a chance to stay out of the cold. The earth itself has also produced an internal source of heat: its core. The core of the earth is 5000 degrees Celsius (9000 degrees F). The best chance to survive the cold is to move to a geothermal place like Iceland and Yellowstone.
Between one and three years, all the oceans of the world will be frozen. Under the thick frozen layer there will most likely still be liquid water. This is because the ice is a good isolator and actually floats on top of the water and the water is heated from below by the core of the earth. This can stay that way for millions of years, allowing tiny life forms to stay alive.
People wouldn't be able to live off vegetable food anymore. Only plant based supplies that have been taken from supermarkets which were still in operation at the time could provide plant based food. When you are at a heat source in Iceland or Yellowstone it is possible to get water, there is plenty of ice or snow. But problems will soon arise. For example, food, you can take animals with you and have them bred to get meat, but many animals live off plants. It will be quite a challenge to constantly supply yourself with meat.
In the best case you keep animals in captivity that are also carnivores. Think for example of a kind of bird that can reproduce quickly and then feed it to a larger carnivorous mammal. You could also breed certain carnivorous arthropods. In any case, it is clear that it will be a heavy diet.
Suppose people don't survive. When the earth shoots out of its orbit, it will travel through space at a speed of 108,000 km per hour. After 1 billion years, the Earth has traveled about 100,000 light-years. It may well be that during its journey through our universe, the earth orbits around a star again and allows the miliscule life forms under the ice to grow and evolve into larger life forms and perhaps even eventually to a human species again.