Bottom-lining it, in The Lord of the Rings "the Wise" is a reference to the White Council, but earlier usage is more general.
When Tolkien refers to "the Wise"1, he's referring to the White Council, also called the Council of the Wise. This is made clear in Appendix B, of all places:
c. T.A 1100 The Wise (the Istari and the chief Eldar) discover that an evil power has made a stronghold at Dol Guldur. It is thought to be one of the Nazgûl.
Return of the King Appendix B "The Tale of Years" (ii) The Third Age
The exact membership of the White Council is unknown, but we know that it included:
- Gandalf (Mithrandir)
- Saruman (Curunír)
From The Silmarillion (emphasis mine):
Ever most vigilant was Mithrandir, and he it was that most doubted the darkness in Mirkwood, for though many deemed that it was wrought by the Ringwraiths, he feared that it was indeed the first shadow of Sauron returning; and he went to Dol Guldur, and the Sorcerer fled from him, and there was a watchful peace for a long while. But at length the Shadow returned and its power increased; and in that time was first made the Council of the Wise that is called the White Council, and therein were Elrond and Galadriel and Círdan, and other lords of the Eldar, and with them were Mithrandir and Curunír. And Curunír (that was Saruman the White) was chosen to be their chief, for he had most studied the devices of Sauron of old. Galadriel indeed had wished that Mithrandir should be the Lead of the Council, and Saruman begrudged them that, for his pride and desire of mastery was grown great; but Mithrandir refused the office, since he would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons. But Saruman now began to study the lore of the Rings of Power, their making and their history.
The Silmarillion IV Of the Rings of Power and the Third Age
The Appendix B quote suggests that Radagast and the Blue Wizards were also part of the Council, and it's interesting that they're not mentioned in The Silmarillion. Of course they don't really factor into much anyway, so when discussing the Council it's usually safe to omit them.
Interestingly, the name "the Wise" was applied by early Men to the people of Finrod Felagund, the lord of Nargothrond in the First Age:
Thus it was that Men called King Felagund, whom they first met of all the Eldar, Nóm, that is Wisdom, in the language of that people, and after him they named his folk Nómin, the Wise. Indeed they believed at first that Felagund was one of the Valar, of whom they had heard rumour that they dwelt far in the West; and this was (some say) the cause of their journeying. But Felagund dwelt among them and taught them true knowledge, and they loved him, and took him for their lord, and were ever after loyal to the house of Finarfin.
The Silmarillion III Quenta Silmarillion Chapter 17: "Of the Coming of Men into the West"
It's not clear from the text whether "Finrod's people" applies only to the Eldar (which would really be the Noldor and Sindar, since that's all the Eldar that Men would have been exposed to), or all Elves, or just the residents of Nargothrond.
Part of History of Middle-earth suggests that this is a reference exclusively to the Noldor:
The Noldor are the Wise, and the Golden, the Valient, the Sword-elves, the Elves of the Earth, the Foes of Melkor, the Skilled of Hand, the Jewel-wrights, the Companions of Men, the Followers of Finwë.
History of Middle-earth X Morgoth's Ring Chapter 2 "The Later Quenta Silmarillion" Part 3 "Of the Coming of the Elves"
There are periodic mentions of "the Wise" being applied to other groups throughout History of Middle-earth, usually with some qualifier ("the Wise of Númenor", for example). Generally speaking, "the Wise" can be used to refer to the body of learned scholars of any civilization.
Initially, "Eldar" referred to all Elves encountered by the Vala Oromë. Over time it narrowed to refer only to the Elves who began the Great Journey, the trip to Aman that not all Elves began:
In the beginning the Elder Children of Ilúvatar were stronger and greater than they have since become; but not more fair, for though the beauty of the Quendi in the days of their youth was beyond all other beauty that Ilúvatar has caused to be, it has not perished, but lives in the West, and sorrow and wisdom have enriched it. And Oromë loved the Quendi, and named them in their own tongue Eldar, the people of the stars; but that name was after borne only by those who followed him upon the westward road.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of the Coming of the Elves and the Captivity of Melkor"
Note that there's no discussion about completing the trip to Aman. The upshot is that, for all practical purposes, "Elves" and "Eldar" are indeed interchangeable; with exactly two exceptions2, every named Elf in the Legendarium is counted among the Eldar. The only Elves who didn't begin the Great Journey, called the Avari ("Unwilling"), just sort of fade out of history. It's possible that they had some dealings with early Men, but we know nothing whatsoever about them as individuals.
1 Note the capital 'W'. There's a distinction to be drawn between "The Wise" as an organization and "the wise" as an epithet. Tolkien was quite careful with his capitalizations, after all, and the list of people who were called "X the wise" is not a short one.
2 Because I just know you're curious, the two exceptions are Morwë and Nurwë, and they're referenced only in History of Middle-earth X; so if you run across their names, chances are good you already care about the distinction between Eldar and Avari