What is "Apologetics", exactly, and how is it used to defend the Christian faith? For those who are unfamiliar with  the term, or those who have only a passing familiarity, what is apologetics, in relation to Christianity?  Is it "proving the existence of God"?

For that matter, can the existence of God be proved via apologetics?  And historically, how has the field of apologetics been used to defend the Faith?
From  http://carm.org/introduction-apologetics


  The word "apologetics" comes from the Greek word "apologia,"
  pronounced "ap-ol-og-ee’-ah." It means, "a verbal defense." It is used
  eight times in the New Testament: Acts 22:1; 25:16; 1 Cor. 9:3; 2 Cor.
  10:5-6; Phil. 1:7; 2 Tim. 4:16, and 1 Pet. 3:15. But it is the last
  verse that is most commonly associated with Christian apologetics.
  
  "....but sanctify Christ as Lord in your hearts, always being ready to
  make a defense to everyone who asks you to give an account for the
  hope that is in you, yet with gentleness and reverence" (1 Pet. 3:15,
  NASB).


Jumping ahead to the second part of the question: "For that matter, can the existence of God be proved via apologetics?" the answer is "no".  

If nothing else, this site, and the questions on it from the atheists and detractors show that the acceptable level of "proof" of God's existence is variable, and even if God Himself were to appear before a non-believer and demonstrate His power, it's quite possible that the non-believer would still refuse to believe that this is God.  To the true skeptic, no proof would be convincing enough. They would always suspect some sort of trickery, like that used by Ardra in the Star Trek TNG episode Devil's Due.

That said, there comes a question of the burden of proof.  Apologetics entails the application of logic and reason to defending the faith, which is quite different than proving the faith.  When it comes to establishing the burden of proof, the same rules that apply in general debates apply to apologetics.  From http://www.nizkor.org/features/fallacies/burden-of-proof.html


  In many situations, one side has the burden of proof resting on it.
  This side is obligated to provide evidence for its position. The claim
  of the other side, the one that does not bear the burden of proof, is
  assumed to be true unless proven otherwise. The difficulty in such
  cases is determining which side, if any, the burden of proof rests on.
  In many cases, settling this issue can be a matter of significant
  debate. In some cases the burden of proof is set by the situation. For
  example, in American law a person is assumed to be innocent until
  proven guilty (hence the burden of proof is on the prosecution). As
  another example, in debate the burden of proof is placed on the
  affirmative team. As a final example, in most cases the burden of
  proof rests on those who claim something exists (such as Bigfoot,
  psychic powers, universals, and sense data). 
  
  Examples of Burden of Proof
  
  
  Bill: "I think that we should invest more money in expanding the interstate system."
  Jill: "I think that would be a bad idea, considering the state of the treasury."
  Bill: "How can anyone be against highway improvements?"
  Bill: "I think that some people have psychic powers."
  Jill: "What is your proof?"
  Bill: "No one has been able to prove that people do not have psychic powers."
  "You cannot prove that God does not exist, so He does."
  


That last one demonstrates the fallacy of thinking that applies apologetics can prove the existence of God.  If the Christian faith rested on the ability to conclusively prove the existence of God, we'd be in trouble for the reasons stated above.

However, apologetics isn't about proving that God exists, or that Christianity is true.  It's about defending the faith against the arguments that say it couldn't be true, logically.  

Put simply, there is, and always has been, a requirement of faith within Christianity. Faith has been a central tenet throughout all of Scripture.  The question comes down to whether the faith is a reasonable faith, or simply a blind, indefensible, fairy-tale faith.  The field of apologetics, as applied to defending the faith from the atheist attacks isn't about proving that Christianity is true, but rather that it is not irrational to believe that it can be true.  Once that is established, we can defend the belief based on historical, archaeological, and logical arguments.

In effect, the field of apologetics accepts the idea that we may be wrong - Christianity may be based on faulty assumptions, but does the evidence lean toward that conclusion, or is does it lean toward the idea that Christianity is Truth?  Can we defend the argument that the Bible is inerrant and infallible?  Can we defend the argument that it's reliable and hasn't been changed over the years to the point where we can't trust it?

Moving back to the first part of the question: What apologetics is....

Also from http://carm.org/introduction-apologetics


  Basically apologetics can be evidential (often called "classical") or
  presuppositional.  Evidential apologetics deals with the evidence for
  Christianity: Jesus’ resurrection, the biblical manuscripts, fulfilled
  prophecy, miracles, etc. Presuppositional apologetics deals with the
  presuppositions of those who oppose Christianity, because
  presuppositions affect how a person views evidence and reason.


Neither can stand on it's own.  Evidential apologetics often deals with unprovable assumptions, but with evidence. 

For example, the death and resurrection of Christ.  This is the central event that defines Christianity.  If this never happened, then Christianity has no basis whatsoever.

Today, right now, we can't go back and prove that Christ died, was buried and was resurrected.  Skeptics have offered plenty of counter-arguments to the Biblical accounts. 

The "Swoon theory", for example, gives us an example of how the two complement each other.  The swoon theory, for those that don't know, is that Christ didn't really die - that He appeared to have died, and then recovered.)  

As evidence for this (evidential/classic apologetics), we have only the evidence from Scripture, and one or two extra-Biblical accounts.  We have historical evidence, but not proof.  The skeptic can easily say "Yeah, the only "historical evidence you have is recorded in your made-up Bible, and a few superstitious extra-Biblical accounts from the time."

To counter this, we fall back on presuppositional apologetics - reasoning logically against the presuppositions of the skeptic.  We counter with all the arguments freely available at http://carm.org/swoon-theory

So, again, apologetics isn't so much about proving the Truth of Christianity as proving that Christianity is not unreasonable.  That it very easily could be true, and added to that, "this is the evidence that convinces us that it is true."

As for the historicity of apologetics, it has not always been only about defending the faith from atheistic beliefs.  It was also used to defend Christianity from fabricated charges.  From http://en.wikipedia.org/wiki/Apologetics#Christian


  Tacitus wrote that Nero fabricated charges that Christians started the
  burning of Rome. Other charges included cannibalism (due to a literal
  interpretation of the Eucharist) and incest (due to early Christian's
  practice of addressing each other as "brother" and "sister"). Saul of
  Tarsus, Justin Martyr, Irenaeus and others often defended Christianity
  against charges that were brought to justify persecution.


In short, the field of apologetics is used primarily to defend the faith against detractors of all types - those who fabricate theories trying to disprove Christianity, those trying to show that it is an "evil" religion, or any other attack.  In general, it's not offensive in nature, trying to force others to believe, but rather defensive, saying "Hey, my belief is reasonable and here's why."  It's not about proving it as providing the believer with the assurance that their faith is not as silly, unreasonable, and evil as opponents would have you believe.