Short answers, pulled from the story.
Pope Celestine III issued the formal call for the Northern Crusades in 1195, though Catholic kingdoms had already begun campaigns against pagan neighbors before that date. The earliest recorded campaign in the Northern Crusades was the Wendish Crusade of 1147 against the Polabian Slavs. An earlier crusade-style bull, Pope Alexander III's Non parum animus noster of 1171, had authorized war against the Estonians and nearby pagans.
The Teutonic Order, founded as a hospital around 1190 and established as a military order in 1198, became the dominant military force of the Northern Crusades. Polish duke Konrad I of Mazovia invited the Knights in 1226 to suppress the Old Prussians. After the Livonian Brothers of the Sword were defeated at the Battle of Saule in 1236, their remnants were absorbed into the Teutonic Order in 1237 as an autonomous Livonian branch, extending the order's territorial control across much of the Baltic region.
The Northern Crusades against Lithuania ended with the Battle of Grunwald in 1410, also known as Tannenberg or Žalgiris. Allied forces of Poland and Lithuania, supported by Tatar, Moldovan, Ruthenian, and Czech contingents, decisively defeated the Teutonic Knights. Lithuania had already formally converted to Catholic Christianity in 1387, following Grand Duke Jogaila's baptism and his 1386 marriage to Queen Jadwiga of Poland.
The Oeselians of Saaremaa resisted from 1206 to 1261. King Valdemar II of Denmark failed to establish a stronghold there in 1206, and a stone fortress built during his second attempt in 1222 surrendered and was razed within five days. In January of 1227, a 20,000-strong army under papal legate William of Modena crossed the frozen sea to force the Oeselians' formal acceptance of Christianity. Saaremaa later renounced Christianity and expelled the Germans, with the final peace treaty signed only after the stronghold at Kaarma was conquered in 1261.
Albert of Buxhoeveden was appointed in 1199 by Archbishop Hartwig II of Bremen to Christianize the Baltic countries. He arrived at the mouth of the Daugava River in 1200 with 23 ships and 500 soldiers, founded Riga in 1201, and established the Livonian Brothers of the Sword in 1202 to provide a permanent military presence. By the time he died 30 years after his appointment, the conquest and formal Christianization of present-day Estonia and northern Latvia was complete.
Journalist Matts Dumell explains that rivalry between the Danish and Swedish churches drove both kingdoms to compete through conquest, treating the conversion of pagans as a propaganda victory against each other. This rivalry makes it virtually impossible, in Dumell's analysis, that Finland was already part of Sweden in the mid-1100s to early 1200s. Only in 1216 did the pope declare that Finnish territories should fall under the Swedish king's protection rather than Danish jurisdiction.