Without a doubt, Sauron was the most powerful being in The Lord of the Rings trilogy. Even without the One Ring, he was a Maia of great power, and Gandalf the White confirmed that no one could have stood up to him in The Two Towers novelization. Powerful as he was, Sauron never would have accomplished his rise to power without his greatest servants, the Nazgûl. They were completely loyal to the Dark Lord, and they inspired the same sense of dread that their master did.
Frodo and the Hobbits experienced the terror of the Nazgûl firsthand. The wraiths perused them all the way from the Shire to Rivendell. Along the way, they confronted the Hobbits on Weathertop and stabbed Frodo. Without the help of Aragorn, it's likely that all hope would have ended then and there. Luckily, Nazgûl didn't get the Ring from Frodo, and he eventually dropped it into Mount Doom. That destroyed Sauron, but it also begs a question: What exactly happened to the Nazgûl after Sauron's demise?
How Sauron Made the Nazgûl
Even before he became evil, Sauron was obsessed with order and perfection. His greatest goal was to control everyone and everything because he thought that he knew the best way to do everything. While he didn't accomplish that as Morgoth's servant, Sauron got another try when he became the ruling Dark Lord. He devised the craft of ring-making as a way to exert control over others.
Midway through the Second Age, Sauron disguised himself as Annatar and infiltrated the Elven kingdom of Eregion. There, he taught Celebrimbor how to make 16 Rings of Power, and their wearers would have been under the Dark Lord's sway. Unfortunately for Sauron, the Elves caught on to his game and tried to hide the rings with their secret three Elven rings. Sauron then destroyed Eregion, stole the rings and redistributed them: nine to Kings of Men and seven to Dwarf Lords.
The nine Men used their rings to claim power and glory in life. They became warriors and sorcerers with exceedingly long lives. Yet, life became unbearable for them because their bodies started to decay and fade. Some 550 years after receiving the rings, there was nothing left of their physical form. Thus, they became wraiths, trapped between life and death with their spirits in thrall to Sauron.
What Happened To the Nazgûl When Sauron Was Defeated
To understand what happened to the Nazgûl when Sauron was defeated, one must go way back to creation. When Eru Ilúvatar designed his children, he made the immortal Elves, and then he made mortal Men. While the Elves were prideful of their immortality, Men received the "Gift of Men." That gift was death. While death may not seem like a gift, it was. It ensured that Men didn't grow weary of the world and its trials like the Elves did. Instead, they lived their lives, died and moved on to a world unknown.
When Sauron made the rings of power and created the Nazgûl, he corrupted the very order of creation -- (like how Morgoth created Dragons). Men were not supposed to live for thousands of years. That's why the Nazgûl's spirits were trapped between life and death -- their bodies had died, but their spirits were chained to the world. That miserable existence finally ended when Sauron's One Ring was destroyed. That damnable piece of jewelry was what kept them chained to Middle-earth, and without it, the Nazgûl's spirits simply died and moved on, like they should have done thousands of years prior.