Sauron, like any of the valar or maiar could choose to embody themselves however they wished. They usually chose forms that reflected their domain and nature, but any of them could choose to appear as a horrible being, if the situation demanded it, though only the servants of evil would end up doing this and eventually they would be locked into these horrible forms, as their nature had become so corrupt that they couldn't transform into anything beautiful.
Sauron was exceptional amongst the servants of the enemy in that he could for very long still choose to embody into a fair form, despite having most definitely become as corrupt as Melkor himself. He would use this to his great benefit in the Second Age until the ability was stripped from him, when Iluvatar destroyed Numenor and Sauron's physical form along with it.
Afterwards Sauron could only embody himself in dread and horrible forms, forever locked as the Dark Lord he had become. Additionally, he had become quite tied to his physical form. After losing the ring, he evidently could not regrow his missing finger, as Gollum told Frodo that the enemy had only four fingers in his other hand. Similarly how Morgoth became permanently scarred and maimed from his duel with Fingolfin and the subsequent rescue of his body by Thorondor the eagle.
Though, maybe after regaining the ring, Sauron could've regrown his missing appendage, but who knows.
This tied to a form is also why the wizards were quite limited in their capabilities and could actually perish if killed.