Short answers, pulled from the story.
Taito released Space Invaders in 1978 as the first arcade game to record a player's score, marking the beginning of digital memory in video games. Before this release, arcade machines were designed to be played and then reset with no mechanism to preserve player progress or achievements.
Pop and Chips released in 1985 for the Super Cassette Vision was the first console game to utilize an AA battery to save progress. This innovation set the stage for the memory cards and hard drives that would follow in the industry.
In Eternal Darkness players can save at almost any time but only if no enemies are present in the room. This restriction adds a layer of tension to the act of saving itself and forces players to weigh the risk of losing progress against the convenience of saving.
Savescumming is a controversial practice where players exploit the system to avoid failure by saving before a critical event and reloading the save if the outcome is unfavorable. Game programmers have developed countermeasures such as checking timestamps to detect rapid save and load cycles or using deterministic seeded pseudorandom number generators to ensure random events occur identically every time a save is loaded.
Nintendo's Virtual Console introduced save states to mainstream consoles in the early 2010s. This technology allows players to save the exact state of a game at any moment regardless of the original game's design by storing the contents of random-access memory to disk.
The Legend of Zelda: Oracle of Seasons and Oracle of Ages actively encourage players to share their progress through password swapping side quests. Games like Grand Theft Auto IV have attempted to prevent the use of saved games made by other users while the Wizardry series and the Fire Emblem franchise allow importing save games from one game to another.