Where do the Rings of Power get their powers from?

Question

I think that the One Ring's power got its power from Sauron - that he basically put some of his power in it.

But where did the other Rings of Power get their powers from?

Answer 1

TL;DR: At least for the Three, and probably the Seven and Nine: from the skill of the Elves that made them

Much about the Rings and the "backstory" of LOTR is explained in "Of the Rings of Power and the Third Age" at the end of the published Silmarillion.

All the Rings except the One were made by the Elves, though they used knowledge given by Sauron.

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.

As for the Three specifically:

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 [...] 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.

So Sauron specifically was not directly involved in the creation of the Three, and their power cannot derive from his as the One's does.

It is never suggested that the Three affect the choices/will of their bearers as the One Ring clearly does all through LOTR, or that Celebrimbor put his "will and strength" into the Three as Sauron did into the One.

The Three are a particularly dramatic example of Elvish "craft" that is called "magic" by mortals.

The introduction to the published Silmarillion ("From a letter by J. R. R. Tolkien to Milton Waldman, 1951") states about this Elvish 'magic':

Their 'magic' is Art, delivered from many of its human limitations: more effortless, more quick, more complete (product, and vision in unflawed correspondence). And its object is Art not Power, sub-creation not domination and tyrannous reforming of Creation.

From that same letter:

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.

On the Three specifically:

The Elves of Eregion made Three supremely beautiful and powerful rings, almost solely of their own imagination, and directed to the preservation of beauty: they did not confer invisibility.

EDIT to add more about the Seven and Nine in response to the OP's request in comments:

From the same letter:

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.

Unfortunately it is not entirely clear what "more directly derived from Sauron" means - do the Seven and Nine actually contain a bit of his being, or does it just mean that the powers are Sauron's design, or derived from Sauron's magic but not his personal essence? The reference to his title as "the Necromancer" may suggest the latter - that the invisibility/wraith-transformation powers come from Sauron's evil magic - but it is not stated exactly.

Of the Third Age and the Rings of Power states that the Three were the most powerful of the Elven-made Rings - "Now these were the Three that had last been made, and they possessed the greatest powers" - which may also argue against the Seven and Nine containing Sauron's actual being/power.