Since this question is related to dragons, and dragons are usually related to medieval fantasy (or later) times and genres, I will include tactics and techniques used by the military (and magic) of those (fictional) times. That means no bombs, jets or tanks. Sorry guys.
To find a the weakness of a dragon, first you must know what kind of dragon you are facing.
Many Asian dragons (Chinese, Indonesian, Japanese, et cetera) have long serpentine bodies, thin arms and legs and feet like the talons of a bird: Mostly associated with water. However, most of these types of dragons are guardians and protectors. They may also fly without wings. To defeat dragons like these I would recommend long-ranged weaponry like bows, crossbows or magic. Perhaps fire would work well against them a majority of the time because of how common they are associated with water. You would want to avoid being circled by one, as they could easily utilize their long muscular bodies to ensnare and crush their opponent.
There are other dangerous abilities one would have to be careful of, such as the Bakunawa's ability to kill people by imagining their death and remote in eye contact.
The Rồng dragons are able to change the weather, and are responsible for crops, so perhaps attacking these dragons would be a bad idea, as they would leave the farms of sovereigns unprotected and harmed.
European dragons (French, English, Greek) are usually a lot larger and are able to breath fire. They are horned and much more lizard-like than their Asian counterparts with their bat-like wings and their huge devastating prehensile tails. Some European dragons have lots of fur and fiery manes that run all down their back, instead, because some dragons are flamboyant and fashionable as all ****. Everyone knows about these bad boys. Just hack away at their feet or aim for the chest. The head and neck are no no's. They stand tall so their heads would be out of reach of melee. Fire magic is out of the question: their scales would be resistant! Arrows and bolts would ricochet off their unbreakable scales! Open up their layer of scales then strike!
Catalan dragons can breath poison as well as fire. They don't use breath mints often. This can happen to people, too. Their breath will rot food and knock a man out cold, so don't try to kiss one.
The Sardinian "Scultone" dragon have the power to kill human beings with its gaze, so don't try The Stare on one of those bad boys.
These are only some of the types of dragons in mythology and folklore. Their character, personality and ability were created through literature and fantasy, and even this is loosely-based, or susceptible to change. Honestly, I wouldn't try to pick a fight with a dragon in the first place. If you can make love to it, bingo. No violence necessary. I hope this helped. It was really just a re-write from some information on Wikipedia.