Wherever I look they are always being called "Good" and "Evil".
Are there any canonical names for each side of the conflict?
For example in World War 1 there was the Triple Alliance and the Triple Entente.
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The War of the Ring was mostly fought between two sides, Tolkien often named these sides Sauron and the Free Peoples of Middle-earth. This sentiment is most clearly covered in Robert Foster's Complete Guide to Middle-earth
War of the Ring - The great war fought at the end of the Third Age between Sauron and the Free Peoples, the subject of Lord of the Rings. In the War Sauron was overthrown for the final time and the One Ring destroyed...
Robert Foster, The Complete Guide to Middle-earth, p. 416, entry "War of the Ring"
Christopher Tolkien, in the Unfinished Tales called the above resource "an admirable work of reference" (UT, p 4).
The term Free Peoples is repeated several times throughout the Lord of the Rings, it's first use was in The Fellowship of the Ring, wherein which Elrond describes the Fellowship as representing the races of the Free Peoples:
‘The Company of the Ring shall be Nine; and the Nine Walkers shall be set against the Nine Riders that are evil. With you and your faithful servant, Gandalf will go; for this shall be his great task, and maybe the end of his labours.
‘For the rest, they shall represent the other Free Peoples of the World: Elves, Dwarves, and Men.
Fellowship of the Ring - Book II, Chapter III: The Ring Goes South
As for the side of evil, most commonly used to refer to them is simply the term "Sauron". While this isn't exactly accurate as Saruman attempted to be a player in the War as well, he was never a powerful force and was disposed of rather quickly. Saruman was also under the sway of Sauron. The use of the term "Sauron" to fight the enemy is likely due to the control he possessed over his armies, as none acted without his command or his thought.