Short answers, pulled from the story.
The core Skywalker bloodline consists of Shmi Skywalker, her son Anakin Skywalker (later Darth Vader), and Anakin's twin children Luke Skywalker and Leia Organa. Leia married Han Solo, and their son Ben Solo (Kylo Ren) is the last blood member. Rey Skywalker adopted the name at the end of The Rise of Skywalker but is not related by blood.
Rey adopted the Skywalker surname to honor the family and to cut ties with her own Palpatine lineage. She chose the name after Ben Solo, the last biological Skywalker, died saving her life, making her the last known person to carry the surname.
Anakin turned to the dark side out of fear of losing his secretly pregnant wife Padmé Amidala. He began having nightmares of her dying in childbirth, and Chancellor Palpatine offered him knowledge of the dark side as a means to prevent her death. After crossing to the dark side, Anakin Force choked Padmé on Mustafar; she survived long enough to give birth to Luke and Leia before dying.
Shmi Skywalker, portrayed by Pernilla August, is the mother of Anakin Skywalker and the root of the entire bloodline. She was a slave on Tatooine, owned by the junk merchant Watto, who later freed and married moisture farmer Cliegg Lars. She was kidnapped by Tusken Raiders and died in Anakin's arms after he found her beaten and tortured in one of their encampments.
Solo was not Han's birth surname. An Imperial officer assigned it to him when he applied to join the Imperial Flight Academy on Corellia, where his father had constructed ships. Since his father's surname is not recorded, Han is considered the first Solo.
In the non-canon Legends continuity, the family extended significantly after Return of the Jedi. Leia and Han had twins Jaina and Jacen Solo, introduced in Timothy Zahn's Heir to the Empire in 1991, and a younger son Anakin Solo. Luke married Mara Jade and had a son named Ben Skywalker. The Legacy comic series, set 125 years after Return of the Jedi, features their descendant Cade Skywalker and a Fel dynasty descended from Jaina Solo.