In the Silmarillion, the Silmarils were described as being very powerful, holy jewels.
What form did this power take exactly?
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The Silmarils contained the combined blessings of the Valar, in addition to a significant part of Feanor's strength.
Displays of power in canon:
The light of the Trees was no mere light, it was the physical manifestation of the blessing of the Valar upon the world, bringing beauty, knowledge and power.
Indirect manifestation in LotR:
The Silmarils had no power in and of themselves. What was contained within them was the Light of the Two Trees, which had been killed by the spider Ungoliant at the behest of Melkor near the end of the First Age.
In J. R. R. Tolkien's legendarium, the Two Trees of Valinor are Telperion and Laurelin, the Silver Tree and the Gold that brought light to the Land of the Valar in ancient times. They were destroyed by Ungoliant at Melkor's behest, but their last flower and fruit were made by the Valar into the Moon and the Sun.
Made by the Noldor, Fëanor, it was the light from the Two Trees that made them significant, no other construct could ever replicate the light from the Two Trees, once they died. The Sun and Moon were sung into existence from the last fruit of the two dying Trees.
What made them worthy of song were the struggles to claim them by all parties involved.
The Valar could have potentially returned the Two trees to life since the Silmarils contained their Light, but Fëanor refused to give them up.
Morgoth wanted them because they were the last, best creation of the First Age and holding the Light of the Two Trees was a treasure of incalculable wealth.
Fëanor's pride in the unmatched quality of his craftsmanship made him unable to part with what was arguably the most famous artifact ever made.
Even the combined skills and ability of all the Valar were unable to replicate their creation. Fëanor's work was unable to be duplicated. Thus the value of the Silmarils was their unique craftsmanship and the primal Light held within.
They would eventually burn Morgoth if he touched them unprotected as he became more corrupt in his nature. Morgoth would eventually lay claim to them and the War of the Valar would be to reclaim them.
Thaddeus has the right answer.
I will add, though, that power (especially magical power) is a very nebulous thing in Tolkien's world. Gandalf does a remarkable job (for a wizard) in using hardly any magic at all; he lights up his staff and makes fireworks. Galadriel, arguably the most powerful or second most powerful non-Ainur in the Third Age, doesn't really exhibit any powers (other than a little fortune-telling). But they both had power to stir people to do great deeds.
And so it was with many things like the Silmarils. They had Power, but they didn't have "powers". Power was an intangible, ethereal device in Tolkien's World (IMHO), and should not be taken too literally.
First of all the power was in them to revive the Two Trees, if they were to be broken so the light would "return to its source" so to speak and second:
Then Elwing and the people of Sirion would not yield the jewel which Beren had won and Luthien had worn, and for which Dior the fair was slain; and least of all while Earendil their lord was on the sea, for it seemed to them that in the Silmaril lay the healing and the blessing that had come upon their houses and their ships. And so there came to pass the last and cruellest of the slayings of Elf by Elf; and that was the third of the great wrongs achieved by the accursed oath. (Silm, Chapter 24, Of the Voyage of Eärendil and the War of Wrath).
The light of Two Trees is in itself great power, each elf who have seen it is stronger and more powerful than others of his kind who do not:
The Noldor, outnumbered and taken at unawares, were yet swiftly victorious; for the light of Aman was not yet dimmed in their eyes, and they were strong and swift, and deadly in anger, and their swords were long and terrible. The Orcs fled before them, and they were driven forth from Mithrim with great slaughter, and hunted over the Mountains of Shadow into the great plain of Ard-galen, that lay northward of Dorthonion.
So maybe Silmarils could somehow enhance natural traits of beings who had them (sort of like the Rings of Power, but in their case through the holy light which the jewels contained), we know that Luthien's beauty was increased (and I doubt that it was only the matter of stylish jewellery:)
But the wise have said that the flame of Luthien wearing the necklace was so great that it was too bright for mortal lands.
This would also explain Carcharoth's passing through Girdle of Melian, also unstoppable rage caused by unbearable pain from holy light and enhancing native power, added significantly to his might, enough to fulfill the prophecy and kill Huan.
Reading from the various examples across canon about the Silmaril, what I see closest describes their inherent power is the power of life and being itself. Vitality (which takes the form of physical strength, longevity, mental clarity and willpower). Having the Light of the Two Trees ingrained in your being makes you more "alive" and stronger in every way which a living being can be strong, be it mental, physical, or magical. Perhaps the Silmaril are simply crystallized ultra-adrenaline :)
Morgoth actually encouraged Ungoliant to suck the Two Trees dry and kill their light, so he didn't value that though they too had been hallowed.
Yet he desired the Silmarils with an unquenchable and jealous desire. Why?
Recall that before Ea was made, Melkor sought in vain "the Flame Imperishable, the Secret Fire", not realizing that it was Eru's alone.
Feanor had a skill of hand and mind that, as Gandalf avers, was "unimaginable" to the Ainur.
I suggest that Eru, for His inscrutable reasons, gave a touch of the Flame Imperishable to Feanor when He caused him to be created.
And that when Feanor imbued the Silmarils with some of his essence, he transferred this Flame to them. He therefore could no longer create their like.
This awoke in Morgoth his ancient longing from since before the World began.