Short answers, pulled from the story.
Trajan's Dacian Wars were fought in two phases: the First Dacian War ran from 101 to 102 AD, and the Second Dacian War from 105 to 106 AD. Both campaigns were fought against the Dacian king Decebalus during the reign of Emperor Trajan.
Rome attacked Dacia for several reasons: Dacian king Decebalus had violated a peace treaty made under Emperor Domitian in 89 AD, using Roman subsidies and engineers to strengthen Dacian defenses instead. Dacia also possessed rich gold and silver mines Rome needed, and Dacia could field an estimated 250,000 combatants, posing a direct military threat.
Apollodorus of Damascus was Trajan's chief military engineer. He built a pontoon bridge across the Danube for the first war, cut a road through the river cliffs, and designed the great stone bridge at Drobeta used during the second war. He is also generally associated with the construction of Trajan's Column in Rome.
Roman forces besieged Sarmizegetusa in 106 AD with the legions II Adiutrix and IV Flavia Felix among the attacking units. After the Dacians repelled the first assault, a treacherous local nobleman revealed the location of the city's water pipes. The Romans destroyed them, cutting off the city's water and food supply, and Sarmizegetusa fell and was razed.
Decebalus fled when Sarmizegetusa fell in 106 AD. Roman cavalry pursued him, and rather than be captured, he took his own life. His confidant Bicilis then revealed to Roman forces where Decebalus had hidden his treasury in the river Sargesia, a hoard estimated by the historian Carcopino at 165,500 kilograms of gold and 330,000 kilograms of silver.
Rome annexed the southern half of Dacia as a new province and settled it with colonists from across the empire. One hundred thousand male slaves were sent to Rome, and legions XIII Gemina and V Macedonica were permanently stationed in the province. The conquered territory went on to contribute an estimated 700 million denarii per year to the Roman economy. Of 3,000 personal names identified in the province, only 60 were of Dacian origin while 2,200 were Roman.