Why were the Rings of Power made, and what were their Powers?

The reason is tied to the regret the Elves had for the passage of time. The Elves were immortal and were fated to live as long as Middle-earth lasted. As such, the earth changed with the passage of time, and the Elves saw many things that were fair become destroyed and lost by the hurts of evil. Sauron, as tempter, awoke a desire in the hearts of Elves to heal the hurts of the earth and create a paradise on this side of the sea to compare to Valinor -- and to be its rulers; whereas in Valinor they were only subjects and below the Valar. The Rings of Power were primarily made to slow the passage of time and preserve their creations of beauty. Yet they had other powers as well.

Tolkien provides a revealing insight on to the nature of the Rings and their powers in one of his letters:

"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... such as rendering invisible the material body, and making things of the invisible world visible."
[The Letters of J.R.R. Tolkien #131]

The Rings were not made as instruments of war or domination; they could not create lightning bolts or hail storms. Yet, they conferred powers commensurate with that of the user; a Great Ring in the hands of a weak and lesser person could not work effects to the extent of the wise or great. Notice Galadriel's words to Frodo in Lothlórien:

"Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others." [The Fellowship of the Ring]

The Elves used the Three Rings to create "islands of timeless beauty" and guard them against the passage of time and evil. Their use can be seen at work at various points:

~ Elrond used the power of his ring, Vilya, to cause the flood of the river Bruinen when the Nazgûl tried to capture Frodo.

~ Galadriel used the power of her ring, Nenya, to keep a guard on Lothlórien so that none could enter without her leave.

~ Gandalf used the power of his ring, Narya, to kindle the hearts and spirits of the enemies of Sauron to do great deeds. And quite possibly to assist him in his battle to the death with the Balrog.

But the use of the Elven Rings was possible only after Sauron was defeated in the Second Age and his Ring taken and assumed lost. If Sauron regained the One, then all the works of the Elves and the use of their Rings would be subject to the evil will of Sauron.