Apologetics sounds like apologize, but actually apologetics comes from a Greek word apologia meaning "a speak in defense". When someone says "Christian apologetics" they most often mean defending the faith or belief in Jesus Christ and the bible. This includes everything from historical facts about Jesus to studies in science to confirm creation.
In biology, mutations are changes to the nucleotide sequence of the DNA in a organism. Mutations can be caused by copying errors in the genetic material during cell division, or other factors like radiation. These "mistakes don't produce new information in the DNA they only damage the original information. Also your body has a built in repair system for fixing DNA Mutations when they occur.
This is basically a principle in biology that says living things that are "more fit" tend to live longer and thus produce more kids then "less fit" animals. For example if a puppy has a Mutation (what) that caused it's legs to be deformed it will probably die sooner then one with normal legs. That dog is more likely to not survive in natural because he is unable to hunt, work, and escape danger.
Carbon 14 is a radioactive element. Radioactive simply means it is unstable and breaks apart over time. This "breaking apart" happens today at a know rate. Because we know has fast (or slowly) it breaks down scientist use that rate somewhat like a clock to measure how long something has been dead.
Macro evolution is a term used to describe major changes with an animal. An example of Macro-evolution would be birds evolving from dinosaurs. For this to happen a dinosaurs would have to develop feathers of which they don't have the information (within their DNA) to produce feathers Micro Evolution would be a term used to describe minor changes that don't involve getting new information. Examples of this would be a wolf changing over hundreds or thousands of years into a bulldog. They are still the same kind of animal, and still interfertile. However their is enough variety within the different kinds of animals that in rare cases the animals changed so much they were no longer interfertile.
To repent means to turn away. When the bible says repent and put your faith in Jesus it means to feel true remorse for your sins (wrong doings) and set out with the intent from then on to never do those things again. If you steal from someone. You would repent by apologizing for doing them wrong and truly never wanting to do that again.
this is a principle used by many scientist today that basically means the way a process happens today is the way it has always happened. In other words if a river dumps 10 rocks a year into the ocean then it has always done that. This concept is used to measure "geologic time". This also eliminates the possibly of a one time catastrophe or any special historical event.
A logical fallacy is basically an error of reasoning. When someone adopts a position, or tries to persuade someone else to adopt a position, based on a bad piece of reasoning, they commit a fallacy. An example would be "Begging the question" also called "circular reasoning" This is when your statement or "assumed facts" assumes your conclusion. Evolutionist use this in there dating methods. Question: How old is this rock? Answer: well it has an "old" fossil in it. Question: How old is the fossil? Answer: It comes from an old rock? This is literally the method used to date fossils. As you can see nothing is proved.