The First Crusade was one of the most remarkable episodes in history. All over Europe, over 100,000 people of multiple nationalities - out of a total population that probably numbered no more than 30 million - spontaneously decided to leave their homes and families, walk 5,000 kilometres through mountains and deserts, and fight a war from which few of them would ever return. 

Later Crusades would be organised by governments, and were better-planned, and often went by sea instead of walking. The other answers to this question have already discussed the mundane and secular reasons for such campaigns - geopolitics, the desire for land, for political status, to find a distraction for troublesome warriors. All of these apply to a greater or lesser degree. However the First Crusade can only be explained as a mass popular outbreak of religious fervour, unplanned and unexpected. 

In about the year 950, the Seljuk tribe of Turkic nomads from Central Asia converted to Islam. They then swept across Iran as conquerors, heading west. In 1055 they captured Baghdad. In 1064, their new leader Alp Arslan invaded and conquered Georgia and Armenia. In 1071 he defeated the Roman army at the Battle of Manzikert, overrunning most of Anatolia and threatening Christian rule in Constantinople itself. His son would expand the Seljuk empire further, capturing Syria and Palestine and triggering a long-running war with the Fatimid dynasty who ruled Egypt.  

The Seljuk invasion triggered a long period of instability and chaos in the Middle East. This was obviously bad for the inhabitants, but it also had negative effects on the Christians who were accustomed to make pilgrimages in large numbers to the Holy Places in Palestine - Jerusalem, Bethlehem, Nazareth and others. The previous Arab Muslim rulers of the region had welcomed these free-spending Christian visitors and protected them. The Turkish Seljuks, however, with the fervour of recent converts to Islam, were much more intolerant. Pilgrims were harassed and persecuted. 

It's unclear how much the people back in Europe were aware of these political changes in the Middle East. From their perspective, it could simply be that the 'Saracens' - their collective name for Muslims of all races - had suddenly decided to start persecuting Christians for obscure reasons of their own. 

For Emperor Alexius I in Constantinople, the Seljuks didn't only threaten a few pilgrims, but the very existence of his empire. In March 1095 he sent a message to Pope Urban II in Rome asking for help. It's likely that what he was expecting to receive was a group of Italian or French mercenary warriors or volunteers to serve in the Byzantine army. He certainly didn't expect 130,000 religious fanatics under no control but that of their own elected leaders... 

It's possible that the Pope had already considered the idea of summoning Christian warriors to go and fight the Saracens, and so Emperor Alexius's request came at an opportune time for him. He embarked on a year-long tour of Europe, primarily in France, preaching in a theatrical manner to large audiences. The response was immediate and huge; many in the crowd swore vows there and then to go to liberate Jerusalem. As news spread, many people in places the Pope hadn't visited - in southern Italy, England, Germany and Spain - also swore to go on Crusade.

"This royal city (Jerusalem), situated at the centre of the world, is now held captive by His enemies, and is in subjection to those who do not know God, to the worship of the heathens. She seeks and desires to be liberated, and implores you to come to her aid. Undertake this journey for the remission of your sins, with the assurance of the imperishable glory of the kingdom of heaven!"

They began setting off in 1096, in waves and small groups, gradually coalescing into a torrent of people flowing down the Danube valley towards Constantinople. Many didn't get there; they got lost, or distracted, or starved to death, or were defeated in battle, or sold into slavery. The whole affair was chaotic, spontaneous, unplanned. By June 1097 about 40,000 of them had gathered into a single large group in Constantinople. The Emperor, horrified at what he'd unleashed, sent them to attack the Turkish-held city of Nicaea. They captured it for him, but then ignored the Emperor's further orders and set off for their true goal, Jerusalem.  

By the time they reached Jerusalem in June 1099 there were no more than 13,000 or so left, barely 10% of those who'd started out. Against all the odds, they captured the city - then massacred many of its Muslim and Jewish inhabitants, in order to - as they saw it - punish them for their blasphemies against God. The Crusaders' success was only possible because the Middle East was torn by conflict between Seljuks and Fatimids, so opposition to them was fragmentary and uncoordinated; but to the Crusaders themselves, who were well aware of how outnumbered they were, it was clearly a miracle due to Divine intervention. 

What motivated them to leave home and travel so far, on a journey on which over 90% of them would die or be enslaved? Certainly not greed. Crusading was expensive; many were forced to sell or mortgage their land, or take out loans from (Jewish) moneylenders. Others relied on support from other family members. Later on, after crusades became more organised, governments levied taxes and the Church organised charitable fund-raising appeals by which people who couldn't crusade themselves could donate money to those who could. For the First Crusade, though, such financial assistance wasn't available. 

The search for adventure, or the desire to escape the sheriff's officers or a girl's angry father, might well have motivated some. Crusaders also had legal privileges; for example, they didn't have to repay any debts they owed until after they returned from crusade. But for most - at least for the First Crusade - religion seems the main motivating factor.  

The idea of pilgrimage was already firmly established in Christian doctrine, and was popular. Rich and poor alike went on pilgrimages; the wealthy simply travelled in more style and perhaps went further afield. A pilgrimage was an act of devotion; visiting a place where holy people had lived and acted was considered meritorious in itself, and an opportunity to reflect on their works. More importantly, it was seen as an act of penance. The hardships and expense of travel, done in such a worthy cause, were seen as a way in which someone could earn absolution from their sins. 

A crusade was regarded as an armed pilgrimage. Crusaders took a vow similar to that taken by pilgrimages, and were covered by the same laws and customs. Mediaeval sources frequently use the word pilgrims (peregrinati) to describe groups of crusaders. As such, a crusader received the same spiritual benefit as a pilgrim; their participation in the crusade was an act of penance, in return for which they received an indulgence absolving them of the penalties for their sins. 

However, more important than that was religion as a form of tribalism. Christians in the Holy Land were under attack by infidels, and so it was the duty of other Christians to come to their aid. Moreover, the Holy Land was God's own home on earth, the place where he'd taken human form, and for it to be ruled over by unbelievers was seen as a disgrace and a source of shame. 

This was the age before nationalism, and most people did not feel patriotism to anything as abstract or remote as their country. Loyalty was a personal thing, owed to your lord or king as an individual. As a nobleman or knight, you knelt in front of your lord and put your hands between his, and swore to follow him and obey him with your very life; and in return he swore to protect you and treat you justly. This ritual of homage was seen as the glue that held all of society together; and it meant that when your lord was harmed in any way it was your moral and legal duty to help him recover his rights. 

What's the relevance to the crusades? Because Jesus was Lord, and his father was the Lord God. Nowadays we might see that as just an expression, a traditional title of respect; but to mediaeval nobles, the word 'lord' (dominus, seigneur) had a very specific legal and even emotional meaning. The Saracens had conquered the Holy Land, and the preachers of the First Crusade compared that quite explicitly to a foreign invader invading a human lord's estates and taking his demesne for their own.  

That was an injury that every nobleman and knight knew that it was a vassal's duty to avenge - and all Christians were Christ's vassals.