Would the world freeze up with two month long nights? Would it burn in the day? How would animal life adapt to such long cycles? Plant life?
Assuming that the planet is in the right orbit (not to close to the star)...
The radiation that the Earth receives from the Sun is strongest at the equator. The air in this region is heated until it rises up through the atmosphere and heads towards the poles of the planet where it subsequently cools. This cool air falls through the atmosphere and is ushered back towards the equator. This process of atmospheric circulation is known as a Hadley cell.
If a planet is rotating rapidly, the Hadley cells are confined to low latitudes and they are arranged into different bands that encircle the planet. Clouds become prominent at tropical regions, which are important for reflecting a proportion of the light back into space.
If, on the other hand, the planet is a slow rotator, then the Hadley cells can expand to encompass the entire world. This is because the atmospheric circulation is enhanced due to the difference in temperature between the day and night side of the planet. The days and nights are very long, so that the half of the planet that is bathed in light from the star has plenty of time to soak up the Sun. In contrast, the night side of the planet is much cooler, as it has been shaded from the star for some time.
This difference in temperature is large enough to cause the warm air from the day side to flow to the night side, in a similar manner as opening a door on a cold day results in warm air fleeing from a room. The increased circulation causes more clouds to build up over the substellar point, which is the point on the planet where the star would be seen directly overhead, and where radiation is most intense. The clouds over the substellar point then create a shield for the ground below as most of the harmful radiation is reflected away.
The high albedo clouds can allow a planet to remain habitable even at levels of radiation that were previously thought to be too high, so that the inner edge of the habitable zone is pushed much closer to the star.
“Rotation can have a huge effect, and lots of planets that we previously thought were definitely not habitable now can be considered as candidates,” says Dorian Abbot of the University of Chicago, and a co-author on the paper.
As to the other parts of your question, very long days would have the biggest impact on plants. Most would probably have some kind of dormant cycle built in, where as the sun sets the plant goes to sleep to conserve energy, and wakes up at dawn to grow during the long daylight period. Others may have a life cycle where the seed is dormant through the night, germinates at dawn and grows through the day, then produces seeds and dies as the sun sets.
Animals may have similar cycles, with some herbivores hibernating during the night and eating during the day, and some active through the night eating the dormant vegetation and stored food. Some predators might hibernate, and others might thrive in the darkness.
A true world building question! It would be very interesting to see how life would evolve in such a world. I m pretty sure it would find a way.
As mentionned in the other reply nights and days would be longer. Most animals would hibernate instead of sleeping. Plants would adapt to this very long night day cycle by being "dormant" during the night.
Some animals would still be active during the long night filling this niche ecosystem. They would be similar to fish living deep under the sea. They might have huge eyes and carry their own lightsource or not rely on sight at all.
You would definitely have more plants not relying on photosynthesis. So a lot of mushrooms and fungus. Trees would be rare or maybe even not exit.
Ocean life would also be affected as parts of it rely on photosynthesis but to a lesser degree.