In Scripture, we are commanded to honor Christ by "always being prepared to make a defense to anyone who asks you for a reason for the hope that is in you" (I Peter 3:15). Any time a Christian explains to an unbeliever why he believes what he does, then he is fulfilling this command. However, there are also formal responses to this command, leading to the theological school called "apologetics." That doesn't refer to "apologizing," though it comes from the same Greek word. Rather, it simply means "defense," that is, defending what one believes. Or, to use the definition of Theologian John Frame, apologetics is the application of Scripture to unbelief. In Christian apologetics, there are a number of schools of apologetics, but the two predominant schools are the evidentialist (also called "Thomist," after the most famous apologist of this school, St. Thomas Aquinas, though there are many non-Thomist shades, as well), and the presuppositionalist.
The primary difference falls in what the proper basis is for that defense. The evidentialist, as the name suggests, holds that the Christian should use the common grounds of logic and empirical evidence to convince the unbeliever of Christian truth. The presuppositionalist holds that one must presuppose the truth of the Bible and the triune God in order for logic and proofs to have a rational basis. Thus, the presuppositionalist denies that a believer and an unbeliever have common ground from which to work toward God's truth. For a more-detailed account of the differences, one can look here and here.
I hold to the presuppositionalist approach. As I have noted before, the Apostles never tried to prove the existence of God or the truth of the Bible.When the evidentialist claims to seek "neutral ground," i. e., not bringing any assumptions apart from those shared by the unbeliever, then he is endorsing the autonomy of the unbeliever. In other words, the supposed "neautral ground" claimed by the evidentialist is no such thing, but rather to argue from the assumption of autonomy that is the fundamental error of the unbeliever. The "neutral ground" is actually the ground of unbelief. How can a Christian apologist argue his case from the basis of unbelief?
In addition to Paul's sermon on the Areopagus (Acts 17:16-34), I look at his words in I Corinthians 2:1-5: "And I, when I came to you, brothers, did not come proclaiming to you the testimony of God with lofty speech or wisdom. For I decided to know nothing among you except Jesus Christ and Him crucified. And I was with you in weakness and in fear and much trembling, and my speech and my message were not in plausible words of wisdom, but in demonstration of the Spirit and of power, so that your faith might not rest in the wisdom of men but in the power of God." Notice how he describes his evangelistic ministry among the Corinthians: he declared Christ crucified; he did not try to appeal to their natural wisdom with logical arguments and secular proofs. I also consider the words of the Apostle John, in I John 5:9-11: "If
we receive the testimony of men, the testimony of God is greater, for
this is the testimony of God that He has borne concerning His Son. Whoever
believes in the Son of God has the testimony in himself. Whoever does
not believe God has made Him a liar, because he has not believed in the
testimony that God has borne concerning His Son. And this is the testimony, that God gave us eternal life, and this life is in His Son." Unbelief is a lie, not because it doesn't understand the argument from first cause, but because it denies the testimony that God has given of His Son. That is the point of conflict, not any imaginary common ground.
Lastly, consider Paul's words, again, this time from Romans 1:18-22: "The wrath of God is revealed from heaven against all ungodliness and
unrighteousness of men, who by their unrighteousness suppress the truth. For what can be known about God is plain to them, because God has shown it to them. For His invisible attributes, namely, His eternal power and divine nature,
have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. For
although they knew God, they did not honor Him as God or give thanks to Him, but they became futile in their thinking, and their foolish hearts
were darkened. Claiming to be wise, they became fools." Note what he says, that the unbeliever is not ignorant of the truth of the biblical God, but rather has suppressed what he already knows is true. Thus, the error of the evidentialist is in failing to understand the point of conflict: it is not over evidence, but over right versus wrong will. And that must be the point at which the apologist directs the truth of God.