My understanding was that the ring only has real power to Sauron.
For other people who wear the One Ring, they're merely filled with delusions of power. "With me you can conquer nations and even overthrow Sauron himself." The Ring of course doesn't offer this power to the wearer. They're only lies so that the wearer feels compelled to take the Ring to Sauron so that he can have his full power restored.
As for why invisibility, it has two purposes:
In the material world, the user is invisible and so he or she is protected. This helps to ensure that the wearer has the best possible chance of making it to Mordor. The more hands the ring is exchanged with, the lower its chances since it takes time to fill the head of the wearer with lies. This way it keeps the number of ring-bearers to a minimum, maximizing its control.
While the wearer is invisible in the material world, he or she glows brightly in the ethereal world, which is the world the Ring Wraiths can see. When he puts it on, the only people who can see him are Sauron's minions. So teh Bearer is safe from everyone, except the minions of the Dark Lord who are searching for the ring.
Essentially the only true power the ring offers someone is to make things easier for Sauron.
Of course the ring can still be very deadly if a powerful being like Gandalf possesses it, because Gandalf is powerful enough without the ring. Even though the ring won't make him stronger, it will make him greedy, selfish, ambitious, deadly and evil. Gandalf is on the same level of strength as the Balrog (both are angelic beings, though the Balrog is fallen).
That's how I understand it, anyway.