We all know that "almost every creature" who puts on the One ring became invisible and that the creator (Sauron) can use it to control the other rings.
But the other rings:
What are the powers, if there are any, of each ring? Is there a respective user who, like Sauron, has a different or extra power for one specific ring?
Of the Rings of Power and the Third Age is your source for all of this.
Before we begin, it's important to note that there is not actually any difference between the Seven and the Nine. There's absolutely nothing in Tolkien to indicate such a difference, and in fact it seems clear that the effects of them were different based on the species they were given to, rather than anything to do with the Rings themsleves.
In other words, one of the Nine, if given to a Dwarf, would have the same effect on him as one of the Seven. And one of the Seven, if given to a Man, would likewise have the same effect on him as one of the Nine.
So it's actually more correct to speak of the One, the Three and the Sixteen.
Secondly, Tolkien doesn't describe lists of powers like a D&D manual so you're not going to get anything so specific. You also won't get anything such as powers of individual Rings; so far as Tolkien described things, the Sixteen all had the same capabilities, as had the Three (although individual Rings may differ in degree of might, so you'll read about Elrond's Ring being described as "mightiest of the Three", for example).
On to the Rings.
The first set of Rings of Power that the Elves made (I'm excluding the lesser rings Gandalf refers to in Shadow of the Past here) are described as having been made with the following motivation:
"Alas, for the weakness of the great! For a mighty king is Gil-galad, and wise in all lore is Master Elrond, and yet they will not aid me in my labours. Can it be that they do not desire to see other lands become as blissful as their own? But wherefore should Middle-earth remain for ever desolate and dark, whereas the Elves could make it as fair as EressÎa, nay even as Valinor? And since you have not returned thither, as you might, I perceive that you love this Middle-earth, as do I. Is it not then our task to labour together for its enrichment, and for the raising of all the Elven-kindreds that wander here untaught to the height of that power and knowledge which those have who are beyond the Sea?"
It was in Eregion that the counsels of Sauron were most gladly received, for in that land the Noldor desired ever to increase the skill and subtlety of their works. Moreover they were not at peace in their hearts, since they had refused to return into the West, and they desired both to stay in Middle-earth, which indeed they loved, and yet to enjoy the bliss of those that had departed. Therefore they hearkened to Sauron, and they learned of him many things, for his knowledge was great. In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power.
Thus the objective of the Rings of Power was to assist the Elves in making Middle-earth as fair as Valinor. This is described in various places (e.g Letter 131) as a second fall of the Elves; they wished to have the bliss of Valinor but yet remain in Middle-earth as the greatest of the incarnates. Letter 131 further goes on to make it explicit how the Rings would achieve this objective:
The chief power (of all the rings alike) was the prevention or slowing of decay (i.e. 'change' viewed as a regrettable thing), the preservation of what is desired or loved, or its semblance – this is more or less an Elvish motive. But also they enhanced the natural powers of a possessor – thus approaching 'magic', a motive easily corruptible into evil, a lust for domination.
But of course there is a dark side:
And finally they had other powers, more directly derived from Sauron ('the Necromancer': so he is called as he casts a fleeting shadow and presage on the pages of The Hobbit): such as rendering invisible the material body, and making things of the invisible world visible.
The Three Rings of the Elves are called out in Rings of Power as having additional capabilities:
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.
If you're observant you'll have noticed something odd: Tolkien, in Letter 131, notes that prevention of decay was a power that all the Rings had, but yet in Rings of Power he states that the Three were particularly desirable on account of this ability. But how could the Three be so desirable if all the Rings had this power anyway? There's absolutely nothing I'm aware of in his writings that reconciles these two statements, so we'll just have to live with it.
And that's the extent of what we know about the powers of the Rings based on Tolkien's writings. I haven't given every single quote (of course), but the other mentions of them don't contain any further information.
Tolkien does list the powers of the One Ring, hidden in plain sight. Read the poem used in the Ring's creation. Its powers are ruling, finding, summoning and binding. We see these powers acting behind the scenes all over the LOTR. Characters were imbued with a great aura of power, fear and majesty to coerce others into doing their bidding (Sam in the tower). The dark tower was physically held together by the one ring. The Nazgul were bound to life by the one ring.
Characters could hear, see and understand things they couldn't otherwise know about - the finding power, and were shielded from being found out themselves - invisibility. Evil things were drawn to the ring(orc captain, balrog, Gollum) and the ring was drawn to evil things, especially its master. The last thing you ever wanted to do was make a deal with the ringweilder. Isildur unwittingly used its power to curse an entire race of people undead for violating and oath and the ring killed Gollum for breaking his oath to Frodo. The longevity power was accidental.
All the rings of power were made for and by immortals - Elves and Maia. There would have been no need for an immortal to deliberately make a ring to increase his already infinite life span so this power was an accidental effect when a mortal started using something never intended for him/her.
BTW - the other rings were also active behind the scenes in LOTR. During the siege of Minas Tirith everyone's spirits picked up wherever Gandalf showed up - ring of fire. Gandalf used it to heal Théoden. Probably also used it to encourage the council of Elrond to resist Sauron. Gandalf's comment to the Balrog about wielding the secret fire of Anor (the sun) may also have been a reference to the ruby ring. Elrond was a great loremaster and healer due to the ability of the sapphire ring to aid in wisdom and knowledge. Lorien was a manifestation of the power of two magic artifacts - the white ring - to preserve all things unstained and the elfstone, the elessar which was made for Galadriel by Celeborn at her request. That one she gave to Aragorn and he bore its name as a throne title. Gotta read the history of the making of middle earth - the big three volume boat anchor series edited by Christopher Tolkien for this stuff.
Answer to Edlothiad - I have no evidence other than the text itself taken as a whole. There was no specific "J.R.R. wrote to Christopher and said this was the interpretation." But step back and look at the forest instead of the trees. Why would Tolkien put these rings of power in the story and then never have anyone use them? Why did he keep mentioning the Elessar and the green stone and then never have it play a part in the story other than getting made fun of by the Mouth of Sauron? Why did he give seven rings to the Dwarves but there was no correlation between that and their seven kingdoms? Why list the powers of the rings as knowledge and wisdom, rekindling that which was lost and preserving all things unstained and then conveniently provide a loremaster, a place where time seemed not to operate and a wizard really good at rekindling people's hearts to resist Sauron if the reader wasn't supposed to see the connection? If the three rings were really all just the same power, then why wasn't Gandalf the loremaster with time standing still all around him and really good with fire too? The simplest explanation is that the rings had different powers and each of the three weilders used them. I think that the one ring and the three rings were continually active during the whole LOTR and played a major part in driving the plot.
With respect to the sapphire ring "making" Elrond a great loremaster as opposed to his ancestry, I don't think that is the way to look at it. The rings amplified things. You already had to have some idea of how to control people for the one ring to give you the true ruling power. Galadriel, I think, commented that the ring granted power to Frodo according to his stature. So, take Elrond, son of Earendil and fully educated with Elven wisdom and then give him the ring of sapphire and you get a truly amazing individual. Same thing, take Olorin, spending his youth with Lorien and learning compassion and now give him the ring of fire and watch as he was able to turn the tide of history encouraging people to resist. Take a sorcerer and give him one of the nine and he turns into the Witch King - total bad ass. It takes gold to breed gold (and a ring). Take a disinterested hobbit, like Bilbo and the ring doesn't confer much power of control. Take another one with a great need to control a capricious and evil guide (Frodo to Gollum) and the one ring conferred power to coerce and curse.
None of the rings gave you powers you didn't already have a little bit of anyway - they amplified what you had.
