The Carter family — into which Johnny Cash married — is central to the history of country music and, therefore, American music.
These two excerpts sum it up. The first, from Southern Music Network, introduces the characters:
The original Family consisted of Mother Maybelle Addington Carter (1909-1979), who played guitar and sang harmony; Sara Dougherty (1898-1979), who played autoharp and sang alto lead; and Sara’s husband, Alvin Pleasant (A.P.) Carter (d.1960), who played fiddle and sang bass.
In the second, CMT describes why they are remembered:
The most influential group in country music history, the Carter Family switched the emphasis from hillbilly instrumentals to vocals, made scores of their songs part of the standard country music canon, and made a style of guitar playing, “Carter picking,” the dominant technique for decades. Along with Jimmie Rodgers, the Carter Family were among the first country music stars. Comprised of a gaunt, shy gospel quartet member named Alvin P. Carter and two reserved country girls — his wife, Sara, and their sister-in-law, Maybelle — the Carter Family sang a pure, simple harmony that influenced not only the numerous other family groups of the ’30s and the ’40s, but folk, bluegrass, and rock musicians like Woody Guthrie, Bill Monroe, the Kingston Trio, Doc Watson, Bob Dylan, and Emmylou Harris, to mention just a few.