I suppose that the most wide-recognised form of Christian apologetics is evidentialism. As practiced by Josh McDowell (Evidence That Demands a Verdict, etc.) and others, this type of apologetics seeks, as the name implies, to pile up historical, scientific, and others kinds of evidence to prove that Christianity is ‘true’.
Presuppositionalist apologetics, on the other hand, insists that the natural mind is so perverse that it is incapable of being converted by evidence and argument. What is required, according to reformed theologians such as Cornelius van Til and others, is a thorough-going exploration of the unbeliever’s assumptions and presuppositions, along with an exposition of the Christian and biblical world view. The unbelieving world view will, it is expected, be shown to be incoherent, and the Christian world view coherent. The individual is then in a position to call out to God for salvation.
In a word, the evidentialist approach is an appeal to reason; the presuppositionalist approach is an appeal to revelation. Or, to put it yet another way, for the evidentialist God is the conclusion of the argument; whereas for presuppositionalist God is the starting-point.
The two approaches are often assumed to be mutually incompatible. But what does the Bible say? Paul, in 1 Corinthians 2:14, would appear to support presuppositionalism, when it teaches, ‘The man without the Spirit does not accept the things that come from the Spirit of God, for they are foolishness to him, and he cannot understand them, because they are spiritually discerned.’ However, Luke appeals to reason and evidence when he writes (Luke 1:3f), ‘Since I myself have carefully investigated everything from the beginning, it seemed good also to me to write an orderly account for you, most excellent Theophilus, so that you may know the certainty of the things you have been taught.’
It would seem, then, that the Bible itself seems to supports aspects of both approaches to apologetics, while also demonstrating awareness of their limitations.
See the relevant articles in New Dictionary of Christian Apologetics (IVP), esp. ‘Evidentialism’ and ‘Presuppositional apologetics’.