I have not read the Lord of the Rings series, but I love the movies. The only power that I have caught in them is the ability to turn invisible, which Bilibo shows in the FOTR. Throughout the entire movie, I feel like they overrated the rings without actually exploring all of the possible powers derived from it. Was there anything in the books? Were powers of the rest of the rings different from the powers of the One Ring?
Letter 131 gives the fullest account:
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. 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.
What's interesting about this is the note that the preservation aspect of this power was an Elvish motive, and an earlier passage in the same letter elaborates on this motive:
They wanted the peace and bliss and perfect memory of 'The West', and yet to remain on the ordinary earth where their prestige as the highest people, above wild Elves, Dwarves, and Men, was greater than at the bottom of the hierarchy of Valinor.
The power of invisibility is noted to have been derived from Sauron, and the letter notes that the Three Elvish Rings didn't have this ability, and of course these Three Rings are recorded elsewhere (in LotR and the Silmarillion) as having been made without Sauron's influence.
The longevity that the Rings grant is not an intended power, but is something that Gandalf notes (in FotR, Shadow of the Past) as being more of a side-effect that is only experienced by mortals (by which we can assume he specifically means Men, including Hobbits as a sub-species of Men) who use a Ring:
A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings.
The chief purpose of the One Ring is the domination of the other Rings and control over all that is done with them.