"Apologetic" has a specific meaning in religious and philosophical literature. In classical Greek, the word apologia referred to a justification or a defense. This is the sense of the word that's used in the title of Plato's Apology; the main character of the dialogue is not saying he's sorry; rather, he's defending his actions before a court.
In Christian contexts, "apologetic" describes a genre of literature that aims to prove that Christianity is a peaceful, rational, salvific religion. The first audiences of these apologetic works were usually Roman officials who harboured various biases against Christians. (One second-century Christian named Justin Martyr dedicated two apologetic works to the emperor Antoninus Pius, though it's unlikely that the emperor ever laid eyes on them.)
Even after the fourth century, when Christianity became the dominant religion in the Roman Empire, apologetic works continued to be written. Obviously there was less of a sense of urgency, since the "enemies" that the books addressed were no longer in any position to punish anyone. Apologetic writings were sometimes directed to heretics or to random straw men, but just as often, they were a kind of self-congratulatory pep talk, used entirely in-house.
Today, the word "apologetic" is almost always used as an insult in the academic world. An author who tries to write about the Bible or about church history in an objective way will swiftly be accused of writing "apologetic" if she shows too much sympathy for a Christian position. I hasten to add that apologetic-masquerading-as-science is a very serious problem in the study of religion. I've read countless studies of "the Historical Jesus" that try to sneak in an evangelical message through the footnotes, and good research is often compromised by a starry-eyed obsession with the One True Way.
This is not just a problem in the study of the Bible, either. Biographers frequently fall dizzily in love with the figures they study, even (or especially) when the subject of the biography is provocative or controversial. It can be hard to resist the temptation to write one's book in an apologetic or hagiographic vein, especially when one feels like one has to defend one's interest in an obscure or unpopular topic. In these cases, apologetic writing can take exactly the opposite tone from the more well-known meaning of the word "apologetic": rather than shuffling their feet and mumbling "I'm sorry," writers of apologetic in this node's sense can be shrill, defensive, and closed-minded.
All that having been said, not every accusation is justified. I have seen very fine authors tarred as "apologetic" simply because they were less skeptical about Christianity than the reviewer. Biblical minimalists are especially keen to call a book apologetic if it admits, for instance, that King Solomon may have existed, or that Jesus may have been a nice guy. I have even seen non-Christian authors accused of writing Christian apologetic, as if a trace of sympathy for a medieval bishop proves that you've thrown your lot in with the Holy Mother Church.
It's a good idea for readers to be on their guard against hidden apologetic. No scholarly work is ever as objective as it tries to be, and even the best-intentioned authors often find themselves defending a position for personal reasons rather than because the evidence supports it. This is especially the case in the study of religion, though it creeps into literary criticism, philosophy, and political science as well. But at the same time, calling something "apologetic" just because you don't agree with it is poor sportsmanship -- something which is probably even more rampant in the academic world than apologetic itself.