Whatever they are, we know that they have a namesake fairy in The Nutcracker who pretty much runs the show. And we know that visions of them dance in the heads of the sleeping kids in "A Visit from St. Nicholas" (also known as "'Twas the Night Before Christmas").
The most common belief is that they are a confection made from dried fruit. In days gone by, "plum" was a name used for a variety of dried fruits. So these sweet treats were made from dried fruits (prunes, figs, apricots, dates, cherries). The fruit was minced and mixed with honey and often some sort of nut (usually almonds). Then spices were added--things like anise seed, fennel seed, caraway seed and cardamom. They were formed into balls and rolled in sugar or coconut. (You can get Alton Brown's recipe here.)
But historians think that the sugar plums in these Christmas stories weren't made from fruit at all. From the 17th-19th centuries, they were a candy called a comfit, made in a very time-consuming fashion. Samira Kawash writes in The Atlantic:
Confectionery historian Laura Mason calls comfit-making "one of the most difficult and tedious methods in craft confectionery, requiring specialized equipment, careful heat control, and experience." Depending on the size of the finished product, a batch could take several days to complete. Not just anybody could make these candies. Until the advent of machine innovations, comfits or sugar plums were a luxury good, most likely to be found in an aristocrat's pocket or between courses at a banquet.They were called "sugar plums" becasue of the similarity in size and shape to the fruit. But the word "plum" came to be used in a number of other ways. Again, Kawash writes:
Sugar plums, therefore, meant all things sweet and good at the time that Clement Moore wrote his poem and Tchaikovsky composed his ballet. Thus sugar plum visions and the Sugar Plum Fairy continue to dance in our heads--whether we understand it or not!