As mentioned by another answer, albums containing songs that are linked together by a story are one of the more common types of concept album - there's a list of concept albums on Wikipedia which may be worth a browse if you have not seen it already.
A couple that sprung to my mind:
The Pretty Things - S. F. Sorrow
The album is structured as a song cycle, telling the story of the main
character, Sebastian F. Sorrow, from birth through love, war, tragedy,
madness, and the disillusionment of old age. (Wikipedia)
Frank Zappa - Joe's Garage
Zappa self-deprecatingly describes the album as a "stupid little story
about how the government is going to do away with music"... The story
is told by a character identified as the "Central Scrutinizer"
narrating the story of Joe, an average adolescent male, who forms a
garage rock band, has unsatisfying relationships with women, gives all
of his money to a government assisted and insincere religion, explores
sexual activities with appliances, and is imprisoned. After being
released from prison into a dystopian society in which music itself
has been criminalized, he lapses into insanity. (Wikipedia)