In Original Dink, dragons are slightly larger than humans, anthropomorphic creatures with wings, although they cannot fly. Some D-Mods have also added other types of dragons, usually notably larger than the ones in Original Dink. Dragons are encountered in the mindless beast variety, and quite often as intelligent beings as well.
In the original game, dragons are first encountered as a couple who have been keeping people in terror on Joppa Isle. After a short chat with the dragons, a fight ensues and Dink kills them, all the while keeping the townspeople out of harm's way. As soon as the fight is over, one of the townspeople reveals the troubles started when they built their town on the dragons' nest, so the dragons were in fact protecting their own from the invading humans.
Dragons are later encountered in the Darklands, and the secret ice castle. Unlike other monsters in the game, dragons do not have a physical attack, and instead cast a "spell" on the player, which can be highly annoying as it's not possible to avoid getting hit, other than by randomly resisting the magic based on your magic ability (the game rolls a random number between 1 and 40. If this number is lower than the player's magic, the magic is resisted). The dragons on Joppa Isle and the ice castle cast Harm, which does 20 points of damage, and dragons in the Darklands cast Greater Harm, which does 30 points of damage.