We know that the Valar are also called Powers of the World. In The Fellowship, chapter II - The Shadows of the Past, we see Gandalf talking to Frodo:
But there is only one Power in this world that knows all about the Rings and their effects; and as far as I know there is no Power in the world that know all about hobbits.
Is Gandalf referring to the Valar here? If he is, who is this Valar that knows all about the Rings (maybe Aulë)? If he is not, what does he means by "Power"?
The first "Power" Gandalf refers to is most likely Sauron; as someone who was intimately involved in the creation of the Rings of Power, Sauron would be the only one who would know all about them. The second use of the word is tricker.
In most cases, "Power" refers to a particularly great, notably supernatural entity: the Valar are called "the Powers", Morgoth is referred to obliquely as "the Dark Power in the North", Sauron is frequently called "the Power in Mordor/Dol Guldur/Barad-dûr", etc.
However, it doesn't make a lot of sense for him to be discussing the Valar, because there is no single one of them who could reasonably be said to "know all about the Rings." More likely is that Gandalf is using "Power" in a sense similar to the way he often uses the phrase "the Wise": to refer to a specific group of particularly powerful individuals.
Who exactly he includes in that group is hard to say, but if I had to guess:
Sauron, obviously; we get this from Gandalf himself at a few points, as in the following passage:
Though the Stewards deemed that it was a secret kept only by themselves, long ago I guessed that here in the White Tower, one at least of the Seven Seeing Stones was preserved. In the days of his wisdom Denethor did not presume to use it, nor to challenge Sauron, knowing the limits of his own strength. But his wisdom failed; and I fear that as the peril of his realm grew he looked in the Stone and was deceived: far too often, I guess, since Boromir departed. He was too great to be subdued to the will of the Dark Power, he saw nonetheless only those things which that Power permitted him to see.
Return of the King Boox V Chapter 7: "The Pyre of Denethor"
Likely the rest of the Wise. I say likely because of Gandalf's next line after the one quoted in the question (emphasis mine):
[T]here is only one Power in this world that knows all about the Rings and their effects; and as far as I know there is no Power in the world that knows all about hobbits. Among the Wise I am the only one that goes in for hobbit-lore: an obscure branch of knowledge, but full of surprises.
Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"
The "Powers" referred to in this passage are most likely the "movers and shakers" of Middle-earth; the leaders, nations, consuls and individuals who most affect the lives of the people. This would include the more powerful nations, like Gondor, Rohan, Lothlórien, etc, as well as groups like the Wise, and influential individuals like Saruman, Gandalf, Elrond, Círdan, etc. And of course, it includes the biggest power in the world that even the isolated Hobbits have heard about, Sauron. It might also include the Valar, though by this time they had withdrawn from the affairs of Middle-earth (including physically removing the Undying Lands from Middle-earth) and their only obvious involvement was in sending the Wizards.
If Gandalf had said in the 1940's, "But there is only one Power in this world that knows all about nuclear weapons and their effects", he would be saying that of all the powers of the world (Germany, the British Empire, USSR, Japan, the League of Nations, etc), only one power, the USA, understands nukes.
