In 1833, at the age of sixteen, Fanny Alger found herself living with Joseph and Emma as a sort of adopted daughter. Both Joseph and Emma were extremely fond of her, but Joseph loved her as more than a daughter.
Oliver Cowdery and Emma eventually discovered that Joseph's affection for Fanny had become physical and both made their complaints. Emma kicked Fanny out and Oliver accused Joseph of having an affair. Joseph never denied the sexual nature of his relationship with Fanny and insisted that Oliver acknowledge that he (Joseph) refused to call it an affair.
The first time hurts the worst, Emma. You'll get used to it.
Despite no extant evidence, apologists tend to frame the Fanny-Joseph tryst as marriage - A MARRIAGE WITH NO KNOWN OFFICIATOR AND NO WITNESSES. Sounds legit, right? Maybe Jesus himself officiated, or King David, or maybe Solomon (biblical dudes were always visiting Joseph when he needed them.) And yet Emma wasn't so convinced it was a marriage. I wonder why? The LDS Church didn't feel like including Fanny in Joseph's family tree either.
Isn't it a bit strange that Joseph was (kind of) living polygamy in 1833 when no revelation of restoration had been introduced and wouldn't be for several more years? The sealing power wouldn't even be restored until April 1836. Joseph was way ahead of the curve on this one! A better explanation for this whole mess is that Fanny was a tool of Satan sent to tempt Joseph and destroy God's budding church, but, as will always be the case, Satan's plan failed!
Five years later Fanny married Solomon Custer, with whom she had nine children. Unlike her parents, who eventually ended up in Utah, she eventually left Mormonism and joined her local Universalist congregation.
Members of the LDS Church generally know nothing about Fanny and if they do they tend not to bring her up. We're still waiting for the Church to address Joseph's polygamy in their recent series of essays on difficult historical matters.
Here's the Feminist Mormon Housewives podcast episode about her.