What is the prefix in the word apology, and what does it mean?
Random House Dictionary (available at Dictionary.com) gives the etymology of apology as:
1400–50; earlier apologie, late ME apologe (< MF) < LL apologia < Gk; see apologia
This means that the original source of the word was the underived (i.e., no prefixes or suffixes) Greek word apologia. It was then borrowed as a loanword into Late Latin and used in its Greek form apologia. From Latin, the word was next borrowed as a loanword into Middle French in the same form and from there borrowed into late Middle English (around Chaucer's time) as apologe, manifesting the first change in form from the original Greek word. Finally between 1400 and 1450 the first evidence of the English variation of apologie appears in a written source. From this etymology, you can see that apology is not a derived word built from a root and prefix nor a combined word built from Greek or Latin combining forms.
Therefore the prefix apo- is not part of apology. However, also according to Random House Dictionary, the prefix apo- (variation ap-) is a Greek prefix used on verbs to derive nouns like apogee and verbs like apoplexy. The apo- prefix conveys a spatial sense described by the prepositions away or apart. In cases like this, checking for the etymology of a word will tell you whether it is a derived word with prefix and/or suffix or whether it is a root word itself or some other word form. Also, greater acquaintance with root words will expand your knowledge of the etymology of words.