The problem is that you can't really hold her to the same standards that you'd hold a modern pop act to. The music industry wasn't really established in the way we conceive of it now, when she started her career. Her early successes with Sonny & Cher came at the right moment in the zeitgeist and she was really one of the defining female figures of the mid '60s in terms of fashion and contribution to the rumblings that eventually led to the Summer of Love.
When the late '60s counterculture movements left her and Sonny by the wayside, she pulled off one of the first successful reinventions by a female artist, turning away from the hippie folk-rock image she'd cultivated early on to a more sophisticated, glamorous one. This was pretty much unprecedented at the time. She had three insanely big hits in the early seventies (the number one singles "Gypsies, Tramps & Thieves", "Half-Breed" and "Dark Lady") and kept herself in the public consciousness on TV with "The Sonny & Cher Comedy Hour" variety show, which was a huge success. After divorcing Sonny, she ran into kind of a fall in popularity from the late '70s to the early '80s, even though she was still putting out music pretty regularly. She had one Top 10 single, "Take Me Home" in 1979, but other than that, her singles weren't taking the charts by storm. A lot of people thought her career was basically over.
This all changed when she began to take on acting in major film roles during the early '80s. She had toyed with acting a few times in the late '60s, but this was the first time she was taking on serious dramatic roles and gaining major acclaim for them. Her personal life was a bit of a wreck at this point and that led to a lot of tabloids keeping her name in the news. She paused her music career while taking on these acting roles. In 1987, she won the Academy Award for Best Actress. That same year, she made her musical comeback with a self-titled album and got her first Top 10 single in almost a decade. Continuing on this momentum, she released another album in 1989, which had all three of its singles hit the Top 10.
During the early '90s, she contracted chronic fatigue syndrome and had to put a halt to her hectic lifestyle. She began to star in infomercials, as she was no longer physically capable of doing many of the things that had earned her money in the past, like acting and touring. Again, people started to count her out and called it the end of her career. After a few years, she got her physical health a little more in check and was able to record music more regularly and act again. In 1998, she was thrust into the headlines again when her ex-husband Sonny Bono died in a skiing accident. Weirdly, this brought her back into the public consciousness and allowed her to strike with an album that was a huge departure from her previous works: Believe. Notably, the title track is one of the first hit songs to use autotune. Obviously, this was a runaway smash success and gave her a fourth #1 single, a full 24 years since the previous one.
So in summary, Cher is notable because she's been around literally forever and has had more comebacks after "career-ending" incidents than almost any other star in history.