The First Crusade was the most successful crusade. It was called by Pope Urban II in 1095 and got underway the next year. It ended in 1099 with the crusaders having captured much of the Holy Land. The most notable place that the crusaders captured was Jerusalem.
The crusade's original goal was to help the Byzantine Empire. It was under attack by Turks who had defeated it a number of times over the past 30 years or so. The Byzantines asked for help and Pope Urban II decided to call for a crusade. Once the call had gone out, the goal changed from simply helping the Byzantines to actually capturing the Holy Land and returning it to the control of Christians.
This goal was achieved in 1099 (though it did not last). This happened after a number of major battles, notably those at Nicaea, Antioch, Jerusalem, and Ascalon. The most notorious of these today is the siege of Jerusalem and its aftermath where the crusaders massacred the bulk of the inhabitants of the city, Muslim and Jew.