Long-worms were a type of dragon found in the northern parts of Middle-earth, and perhaps elsewhere, generally believed to be long and without legs. The most famous long-worm (and in fact the only one mentioned as such) was Scatha, who preyed on the Dwarves and Men of the Grey Mountains, and was slain by Fram of the Éothéod. But Scatha was assuredly not the only Long-worm, as it says that for a long time after that the northern lands were free from the Long-worms.
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, though he was one of the winged dragons that appeared in the War of Wrath.