After considering the Solas regarding Grace, Faith, and Christ, we come to the heart of the dispute between the Reformers and Rome, the question of authority. How do we know what we know in terms of God and His doctrine? For the Reformers, the very gospel was at stake in how this question is answered.
Roman Catholicism had over the centuries looked to a living and evolving tradition embodied in the Magisterium - the Bishop of Rome (or the Pope), the other bishops, and the doctors of the Church. Utilizing philosophy, tradition, compromise, and innovation, Roman Catholicism emerged over the centuries as a complex body of doctrines and practices, many of which had no basis in Scripture and often represented a distortion of New Testament teachings.
The Reformation is closely connected to the Renaissance of Western Europe, a time wherein thinkers wished to return to the sources (ad fontes), and revisit the centuries of teaching and tradition on many topics - government, philosophy, law, art, and more. The Reformers pursued this course in the context of a Roman Catholic Church which had in recent centuries faced great controversies and division. Returning to the Scriptures anew, they found that Rome's teaching often contradicted the Bible - to the point that Rome had lost the gospel replacing it with something else. This is what inspired Luther to call for reform in 1517. Others picked up the banner and because of Rome's hostility and rejection of their arguments, the campaign of 'protest' eventually resulted in a permanent division.
What are the Scriptures? Christians have always believed they are a revelation from God, indeed the Scriptures speak of the prophets being moved by Holy Spirit (2 Peter 1:21), and Paul argues the Scriptures are Theopneustia or God-breathed (2 Timothy 3:16). The question is do the Scriptures then constitute a sufficient authority (2 Timothy 3:17) for Christian doctrine and life? Rome historically has upheld the divine inspiration of the Scriptures but has treated them (at best) as a kind of starting point on which to build. The Reformers insisted that Rome's approach all but canceled out and undermined the teachings and authority of Scripture (Matthew 15:9) and that Scripture Alone (Sola Scriptura) was the only basis on which to argue and believe.
Roman Catholicism developed the papacy, its ecclesiastical hierarchy, and its many traditions concerning worship and liturgy. Mary evolved into a messiah-like entity worthy of worship. In the Mass, the Lord's Supper when wed to the Roman concept of priesthood became a repeated sacrifice and (through philosophical sleight-of-hand), the doctrine of transubstantiation emerged in the Middle Ages wherein the bread and wine of the Supper become the actual (literal) Body of Christ. Doctrines such as Purgatory would materialize and with it a system of penances and indulgence wherein money and works could be employed to save and to rescue those still suffering unable to enter heaven. The gospel of grace was lost and the Reformers rightly insisted the remedy was to return once again to the source and foundation, the holy writings given by God, the Old and New Testament Scriptures.
This is not to say there is complete agreement or that it's easy. Major questions and debates remain. How do the Old and New Testament relate to one another? Do (as the Lutherans contend) some parts of the New Testament carry more authority than others? There was a long debate over the extent of the Old Testament canon as well as contentions over the New Testament text itself. How is theology to be pursued?
For some, the later addition of Protestant Confessions put limits on the concept of Sola Scriptura. For Lutherans and Anglicans, Sola Scriptura tends to be limited to gospel-related questions. As long as the gospel is not compromised, there is great latitude and thus these bodies have sought to keep as much historical tradition as possible.
Evangelicals pay lip service to Sola Scriptura but demonstrate in practice that it plays little part in how they view the Church, worship, and the Christian life. Disagreements remain, but as Protestants we can (and must) always turn back to Scriptural authority.