You are probably already aware that Sauron is immensely powerful — he's at the same tier as
Gandalf and the Balrog, essentially. He put all his evil and strength in the One Ring, to the point
where he was crippled and relied on the Ring for his strength. This meant that when the Ring was
taken from him and he was defeated, he was left barely alive, forced to exist as the Eye and to seek
out the Ring so he could recover his power. Because he put so much of himself into the Ring, he has
a connection to it, presumably on a spiritual and mental level. The reason why Gollum, Bilbo and
Frodo all struggled to bear the Ring and began to corrupt was that they were exposed to Sauron's
great evil on a level that is beyond mortals; Sauron is, after all, a demi-god who is extremely hard
to kill and has survived since the time of Morgoth.
The power in the Ring was also present in the other Rings, as he was heavily involved in their forging
and subsequent corruption. He linked them to the One Ring and made them conduits for its power.
His influence could reach the Ringbearers through their Rings and send them mad; the Rings were
meant to be tailored to the deepest desires of the wearer. For example, the Rings given to the Men
who would later become the Nazgûl gave long lives because they wanted to live forever. The Rings
gave them what they wanted, but at the price that it sent them mad. Sauron's influence in the Rings
meant that they developed an attachment and they were 'addictive', making the Ringbearers a guaranteed
source of controllable madness at the hands of a demi-god with the remote control, and a constant
source of torment for them through the Rings. They couldn't break away because they needed Sauron
to keep them alive and to keep them from going off the edge.
That's all I can remember off the top of my head, and I'm sure someone who's a better at the lore
than I will come in and correct me. TL;DR: Sauron is basically a drug-dealer for evil.