Before I try to answer this question, let me first say that the term "presuppositional" is not helpful. It is ambiguous and is often thought to mean an affirmation that everyone has presuppositions.
What, then, is "presuppositional" apologetics. In sum, it is the application of Reformed theology to the discipline of apologetics. A better name for this approach could be "Reformed" apologetics, or (my own label) "Covenantal" apologetics. All people are either covenantally in Adam, or covenantally in Christ.
This approach recognizes the practical and methodological necessity of the two principia of Reformed theology. The first, the principium essendi, affirms that the Triune God is the Creator and Sustainer of all that is. This means that He is the Creator and Sustainer of the entire universe and of every person in it. All people, therefore, are responsible to the same Triune God. No one is outside of His sovereign Lordship.
The second principium, the principium cognoscendi, affirms that the foundation for all knowledge is God's revelation — in His Word and in His world. Because God has spoken, it is His speech alone that can truthfully define who we are and what His world is like. God has also spoken in and through His world, including through each person, so that all people know Him truly. This knowledge that all people have, however, is suppressed and held down by those who are outside of Christ, and who remain in Adam.
The central task of apologetics, then, is to discern how those who hold down the truth of their knowledge of God object to this truth, and how we might persuasively answer their objections. Since those objections depend on, even as they suppress, the knowledge of God, we appeal to them as "knowers-but-suppressers" in their objections against, or questions about, Christianity.
K. Scott Oliphint, Ph.D. is professor of Apologetics and Systematic Theology at Westminster Theological Seminary.
For Professor Oliphint's full treatment of this question:
K. Scott Oliphint, Covenental Apologetics: Principles and Practice in Defense of Our Faith.
Professor Oliphint's recommended extra resource on this question:
Cornelius Van Til, A Christian Theory of Knowledge.