Sauron was defeated by Isildur but he did not die. I suppose because the ring was taken from him but it was not destroyed. Having lost the ring how did he begin to gain power again? Did the ring have indeed its own will to help Sauron from a distance?
Sauron was a Maia. A angelic being from before the creation of the universe - The Ring contained and magnified some of his natural power but it wasn't the reason he didn't die. Sauron cannot die, even after his defeat at the end of Lord of the Rings he is not truly dead - merely lacking enough power to do anything.
Incorrect in the question:
Sauron was NOT defeated by Isildur.
Sauron was defeated by Gil-gilad and Elendil, who both died from injuries/exhaustion.
Isildur cut the one ring from Sauron's defeated and broken body with the shards of Narsil.
Answer to the question:
The One Ring is part of a magical means of binding the other users of rings of power to his will (soul). In essence the One Ring is still Sauron, its will is his will. Sauron was weakened enormously by the separation from the ring, and he went into hiding. As the ring houses a great portion of his power or soul (and is undamaged) he was able to very slowly pull himself back together.
Ancalagon answered the other component of the question.