Long-worms were a type of dragon found in the northern parts of Middle-earth, and perhaps elsewhere. The most famous long-worm (and in fact the only one that Tolkien explicitly identifies) was Scatha of the Ered Mithrin, who preyed on the Dwarves and Men of the Grey Mountains, and was slain by Fram of the Éothéod.
Though Tolkien gives almost no clues about long-worms in the text of The Lord of the Rings, his illustrations of dragons give us some further hints. Tolkien's dragons tend to be sinuous, serpentine creatures, having the appearance almost of a winged snake rather than the more traditional dragon-form. This would explain the term 'long-worm' easily. It's interesting to note that Tolkien gave this form to another northern dragon, Smaug, which strongly suggests that he, too, was one of the long-worms.