Dragons are predatory Megabeasts which are extremely hostile, larger than elephants, and quite resistant to damage. In addition to bite and claw attacks, they can breathe extremely hot ( hot) dragonfire over a considerable area, which can cause things such as grass, equipment and people to ignite. This fire is apparently magical in nature, and can hurt many creatures who are otherwise comfortable in the fire. Note that if your Dwarves are wearing shields, they can sometimes block the Dragonfire.
When a dragon enters your area, the game will announce its presence and its name. It will then attempt to eat you. Quickly.
Dragons are building destroyers which means among other things that doors are useless as defense. Even their breath is able to destroy (deconstruct) workshops and turn paved stone roads into dirt.
 Defense strategies
It takes from 3 to 10 stone-fall traps to kill a dragon, though weapon traps loaded with piercing weapons work well too if they get lucky and destroy internal organs. Considering their fire-breathing ability, traps are probably the safest way to manage this creature.
Despite all its advantages, it is possible for relatively small groups of dwarves (even as few as 8) to take down dragons, especially if there are high proportions of relatively well trained marksdwarves available.
Additionally, dragons can be caught in cage traps (Beware, if you embark on a map with an existing Dragon, it will be labeled "local resident" and be immune to Cage Traps) and then tamed or sold to traders. (Beware: if you attempt to transfer a Dragon cage to a trade depot before the dungeon master has arrived, the dragon will escape, wreaking havoc on all your dwarves who are bringing other trade items.)
While the dragon itself is usually fairly easy to deal with, if you are not quick in doing so, it will likely set a large number of objects (and dwarves) on fire. Given that dwarven AI doesn't understand being on fire very well, this can quickly lead to dwarves inadvertently spreading the fire throughout your fortress. Be sure to quickly forbid burning objects that dwarves might want to pick up, and also draft burning dwarves so you can order them to take a bath before spreading the burning horror around.
Timing can make the difference between success and failure. One strategy is to pile a large squad of melee dwarves and marksdwarves on the dragon while it's crossing a bridge / floor-path over a river. If you're lucky, the barrage of bolts will send the dragon falling off of the bridge stunned and you can have your way with it. If done correctly, this is possible to pull-off without a single casualty.
Siege weapons can also work well against dragons -- one shot from a ballista is quite likely to kill or at the very least seriously injure a dragon.
Dragons can be tamed only with the help of the dungeon master noble. Any dragon that has already tasted dwarven blood and received a title for doing so will never be truly tamed. Dwarves will believe the dragon to be tame, and will not run away in fear from it, but the dragon will continue to happily devour them. This bug/feature exists for all creatures that have killed your dwarves, not just dragons.
Even in the case of taming a dragon that has not killed any of your dwarves, beware that dragonfire does not discriminate between friend or foe.
 Irregularities, Bugs, and Planned Features
Note that, as of the current version, Dragons cannot actually fly, which contributes towards the ease of killing them. There are few things more pathetic and commonplace than watching a hapless Dragon trapped at the edge of the map behind a retracted drawbridge and a tiny moat just barely wide enough for two dwarves to slide past one another, slowly succumbing to the target practice of an untrained marksdwarf, who can usually finish him off after one or two trips back to restock ammunition and gain a novice ranking as a reward. Certain dev_notes indicate that in eventual future releases, dragons will not only be able to fly, but will use their flight to a terrifying level of efficiency, strafing your troops with fire from above, as well as picking them up and dropping them from a great height. Unfortunately (fortunately?) these enhancements may not show up in the game for quite some time.
Dragons are apparently immune to dragonfire, but not immune to any other kind of fire. Unfortunately (fortunately?), the dragon AI (like the dwarf AI) seems to believe that it is in fact immune to all fire, and so they will gleefully go swimming in lava-- and hence get burned to a crisp. Their dragonfire can also ignite many objects on the ground, and if the dragon then walks over these objects, they will sometimes ignite themselves in the process. These factors are all likely bugs, and are liable to be fixed at some time in the future.
Not necessarily a bug, but sometimes when a dragon appears in the middle of an ambush or siege, the goblins will attack the dragon, either wounding it greatly or killing it in the process.
[CREATURE:DRAGON] [NAME:dragon:dragons:draconic] [TILE:'D'][COLOR:2:0:0] [MODVALUE:50] [DRAGONFIREBREATH][FIREIMMUNE_SUPER] [FANCIFUL] [MEGABEAST] [GENPOWER:5] [FREQUENCY:5] [PETVALUE:10000] [GRASSTRAMPLE:50] [BONECARN][PET_EXOTIC] [BUILDINGDESTROYER:2] [NOFEAR] [LIKES_FIGHTING] [PREFSTRING:terrible majesty] [BODY:QUADRUPED:TAIL:2EYES:NOSE:2LUNGS:HEART:GUTS:ORGANS:THROAT:NECK:SPINE:BRAIN:MOUTH] [BODYGLOSS:CLAW_FOOT] [SIZE:20] [ATTACK:MAIN:BYTYPE:MOUTH:bite:bites:1:6:GORE][ATTACKFLAG_CANLATCH] [FAT:10] [ALL_ACTIVE] [BIOME:ANY_LAND] [STANDARD_FLESH] [SWIMS_INNATE][SWIM_SPEED:2500] [HOMEOTHERM:10040] [SPHERE:FIRE] [SPHERE:WEALTH] [CURIOUSBEAST_ITEM]