This is my first answer here, and I am only a student of history, so please remove this post if it isn't up to snuff.

The First Crusade largely started as a result of the expansion of the Seljuk Turks into the Levant and Anatolia. The Byzantine Emperor Alexius I exhausted his troops, many of which were foreign mercenaries, fighting the Turks in order to take back Anatolia. Alexius I decided to ask the Catholic Pope, Urban II, to help recruit western Catholic soldiers to serve as mercenaries in the Byzantine campaign against the Turks. As a result Urban II called upon Catholics to help rescue Jerusalem from Muslim occupation, and turn it over to the native Christian population living there. Urban II used the mistreatment of Christian pilgrims to Jerusalem by the Turks in the first few years of their occupation as justification for retaking the city, though by the time the Crusade was called missionaries were largely left unmolested once again. The idea was to help the Byzantines along the way, reclaim Jerusalem for the Christians, and head back home. That's not exactly how it turned out. The declaration of the First Crusade inspired a wave of religious zeal amongst Western European Catholics, and the fight to save Jerusalem quickly took precedence over assisting the Byzantines. While the proper Crusading army was made up of the armies of nobility, the first wave of crusaders to travel east were peasants led by charismatic figues such as Peter the Hermit, who inspired peasants, criminals, low-ranking knights, and other common people to take the cross. This rag-tag army caused all sorts of trouble as they travelled across Europe and Byzantine territory, and ended up getting slaughtered by the Turks in Anatolia. This left a fairly poor impression on the Byzantines. As the "official" Crusading armies travelled through Byzantium they did not behave much better. Raiding of Christian cities in order to procure supplies was fairly common. Moreover, the Catholic nobility did not trust one another to properly cooperate in assisting the Byzantines. They certainly didn't trust the Byzantines. There are reports in the Alexiad of a fair amount of hostility between Alexius I and many of the Crusaders. Furthermore, many key members of the First Crusade were Normans, who had previous bad blood with the Byzantines due to expansion in Sicily and the Balkans. As a result the Crusading Army was unable to cooperate with the Byzantines in order to take on the organized Seljuk Turks. That's why you probably don't hear much about the Byzantine angle on the start of the Crusades.

The Crusaders were able to take Jerusalem due to the fragmented Muslim factions present in the Levant. After traveling through Anatolia they no longer had a strong, centralized force to deal with. Many died of starvation while traveling through the Levant rather than dying from fighting. Upon taking Jerusalem the Western Europeans found the native Christian population unacceptable to take over the city, as was planned. They simply found them and their customs too foreign, and instead decided Western Europeans needed to stay behind in the Levant. This led to the creation of the Crusader States. Following the creation of these Crusader States the Muslim powers in the Levant began to consolidate under a series of powerful leaders. The newly strengthened Muslim powers in the Levant began chipping away at the land claimed by the Crusader States. This would culminate in the retaking of Jerusalem by Saladin, which would trigger the Third Crusade. Following Crusades were largely aimed at reclaiming former Crusader State land lost to the Muslims.

This answer does not cover Crusades in Europe, which are a whole other ball-game. I also gave a rather simplified view of the creation of the Crusader States. I can expand upon these some if requested. Again, this is my first answer, and there are definitely people here more qualified than I am so please remove this if it is not up to quality standards.

Source: The New Concise History of the Crusades by Thomas F. Madden