In short: we first saw the darksaber in the hands of a bad Mandalorian named Pre Vizsla, head of Death Watch and Bo-Katan's one-time boss. He led a coup against Bo-Katan's sister, Satine, who was leading a pacifist government on Mandalore during the Clone Wars. But Vizsla made the mistake of allying with Maul, who pulled off his own coup, killing Vizsla and claiming both the darksaber and Mandalore for himself.
Maul later hid the darksaber away in the old Nightsisters' lair on Dathomir, where a good Mandalorian named Sabine Wren found it during the Rebellion Era. She later gave it to Bo-Katan after they defeated the leader of the Mandalorians who had sided with the Empire. Bo-Katan was supposed to use the darksaber to unite the fractious tribes.
But then the Empire blew up Mandalore, turning its surface to glass and stealing all the beskar. The Mandalorians dispersed throughout the galaxy, some hiding away in "coverts" while others become guerrilla warriors (like Bo-Katan's Nite Owls). It would appear that Bo-Katan lost the darksaber during the Empire's destruction of Mandalore (or thereabouts). It's possible Moff Gideon bested her in combat and claimed it for himself. We don't know yet.
But now she can't just accept it back from Din, even though he's offered it to her, because she hasn't earned it. And it's possible she may never get it back. I still would not be at all surprised if they're building up to Din becoming the new leader of the Mandalorians. Bo-Katan may even end up going back to being a villain again.