Short answers, pulled from the story.
Nut is the ancient Egyptian goddess of the sky, stars, cosmos, and the universe. She is one of the oldest deities in the Egyptian pantheon, with her origins in the creation myth of Heliopolis. She is depicted as a nude woman covered in stars, arching her body over the earth.
Nut's children are Osiris, Set, Isis, and Nephthys. A Graeco-Egyptian version of the myth adds Horus as a fifth child. Plutarch also names Horus the Elder and a figure called Arueris among her offspring.
Nut was painted on the inside lid of sarcophagi to protect the deceased in the afterlife. She was seen as a friend of the dead because of her role in saving Osiris and in the daily rebirth of the sun. Tomb vaults were also painted dark blue with many stars as a representation of her.
The Book of Nut is the modern title for an ancient Egyptian astronomical text originally called The Fundamentals of the Course of the Stars. It dates back at least to 2000 BC and is considered one of the earliest Egyptian astronomical texts. Nut plays the primary role in it because she was the sky goddess.
In De Iside et Osiride, Plutarch describes how the figure he equated with Nut was cursed to give birth on no day of the year. Mercury won five extra days by gambling with the moon-goddess Selene, and these were added to the 360-day calendar as the "Epact" or intercalary days. On each of those five days one of her children was born.
The maqet was a ladder sacred to Nut, representing the ladder Osiris used to enter her heavenly skies. It was placed in tombs to protect the deceased and to invoke the aid of the deity of the dead. The symbol ties Nut directly to funerary practice and the passage into the afterlife.