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 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. 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.
It has been suggested that the Long-worms were original creatures created by Ilúvatar, and that the dragons of Morgoth were bred from them, as something cannot be bred from nothing. This suggestion also implies that the long-worms were witless hunters, like lions or wolves, with the attraction for bright things, like ravens, rather than intelligent beings like the later dragons. It is debatable whether or not Morgoth could create intelligent beings from nonintelligent ones.