Sola Scriptura, or "by Scripture alone", was the main Protestant slogan of the Reformation. It represented the appeal of the Reformers such as Martin Luther and Ulrich Zwingli for people to reject the tradition and inherited practice of the Roman Catholic Church. In 1519 in Leipzig, Martin Luther was faced in debate by Johannes Eck, the champion of the Dominican Order, whom for centuries had staffed the Papal Inquisition. Eck had previously defeated Luther's University superior Andreas von Karlstadt in debate, and he now turned his full power against Luther: he exposed the similarity between Luther's ideas and those of Jan Hus, who had been burnt as a heretic following the Council of Constance in 1415.
Luther now had a choice: repudiate all his ideas or reject the authority of the General Council of the Catholic Church. He did the latter and in doing so formalised the principle of sola scriptura. Protestant thought was now established in thinking not that the tradition of the Church was responsible for interpreting the Scripture, but that the Scripture was responsible for interpreting the tradition of the Church. The Reformers began to reject many practices of the Church which they believed had no basis in Scripture, such as transubstantiation, the veneration of images, the idea the Church could 'store' the Merit of Saints and resell it (indulgences), and the idea that priests were more sacred than the laity.
The Catholic Church had seven sacraments which traditionally guided a person through their life from birth to death. There was ordination (setting a priest aside from the laity), marriage, baptism, confirmation (bringing someone into adult membership of the Church), the "last rites", penance ("confessing") and the Eucharist. Luther could find basis in Scripture for only three - penance, the Eucharist and baptism - and later he removed penance. The new doctrine of the Eucharist was also very polemic, and became known as consubstantiation. The theory of transubstantiation said that when the Eucharist (the consecrating and consuming of the bread and wine) occured the bread and wine actually became the Body and Blood of Christ. Consubstantiation said that the spirit of Christ appeared in it briefly as heat appeared in a hot coal. Later Reformers would see this as merely more superstitious nonsense like what the Church had been accused of.
By rejecting the authority of the Pope and the Catholic Church structure to interpret Scripture Luther had destroyed a lot of the underpinning of the Catholic faith and of society. It was the push towards sola scriptura which led to so many people translating the Vulgate (the official Latin Bible) into local vernacular. This allowed the Word to be read and interpreted by all, which would lead to some madmen claiming to be shown extreme interpretations by God (and in some cases the newfound freedom led to them rejecting the Bible altogether and claiming to have an inner light, as some Anabaptists did). But in general this could only be a good thing - the Church gradually lost its grip and its power. The Church's ability to exploit people was reduced and a greater understanding of the Word was spread and preached, by which people became better educated.
At the Council of Trent, the Catholic Church rejected sola scriptura. It said that tradition was to be relevant as well, and that when Scripture was needed the Vulgate would be consulted. The exact text of the Vulgate was approved by the Pope and the Council and no-one outside of the Church structure was allowed to contribute (unlike the 'unofficial' vernacular editions).