So, I've been reading Lord of The Rings, and other Tolkien books, and I've always wondered something. What did the other rings of power, given to the various races by Sauron actually do?
This is covered in Of the Rings of Power and the Third Age in the published Silmarillion.
Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had, as it seemed, unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men; but too often they beheld only the phantoms and delusions of Sauron. And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore and under the domination of the One, which was Sauron's. And they became for ever invisible save to him that wore the Ruling Ring, and they entered into the realm of shadows. The Nazgûl were they, the Ringwraiths, the Enemy's most terrible servants; darkness went with them, and they cried with the voices of death.
The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows. They used their rings only for the getting of wealth; but wrath and an over-mastering greed of gold were kindled in their hearts, of which evil enough after came to the profit of Sauron. It is said that the foundation of each of the Seven Hoards of the Dwarf-kings of old was a golden ring; but all those hoards long ago were plundered and the Dragons devoured them, and of the Seven Rings some were consumed in fire and some Sauron recovered.
The Elven rings were a special case, as Sauron didn't give those to the Elves; instead, Sauron wasn't even involved in their making, and the Elves had managed to protect them so that he never even touched them. Nonetheless:
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya, and Vilya, they were named, the Rings of Fire, and of Water, and of Air, set with ruby and adamant and sapphire; and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world. But Sauron could not discover them, for they were given into the hands of the Wise, who concealed them and never again used them openly while Sauron kept the Ruling Ring. Therefore the Three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.
And finally, as a general property of all of the Rings of Power:
And all those rings that he governed he perverted, the more easily since he had a part in their making, and they were accursed, and they betrayed in the end all those that used them.
The Elven rings did not make their wearers invisible, but the rings themselves could be so.
The Dwarven rings apparently did not, but the Human rings definitely did. The Nazgul were invisible except for their black cloaks and armor and weapons.