Speaking as Moved by the Holy Ghost
Peter described what prophecies of old time were like, holy men spake as they were moved by the Holy Ghost. The New Testament uses the same word for 'prophecy' to describe what Old Testament prophets did that it uses to describe prophecy in the church. So we should not assume the word has been redefined.
'Speaking as moved by the Holy Ghost' is a good rule-of-thumb definition. The Bible also mentions prophesying on musical instruments and shows prophets doing what some have called 'sign acts', where an act conveys, like wearing a yoke, a dirty old belt, or walking around naked with buttocks bared, conveys part of a prophetic message. It may be that some prophecies are sung, if we consider the Psalms. Peter called David a prophet.
Quoting God
A typical 'format' in which prophecy is conveyed is as a first person quote from God. We see this throughout the writings of the prophets. The prophet might start with 'Thus saith the LORD.' In the New Testament, Agabus starts a prophecy with 'Thus saith the Holy Ghost.' So we should not consider prophesying to be something completely different in the church from what it was in the Old Testament.
A Distinction between Prophesying, Teaching, and Evangelism
In Romans 12, Paul lists prophesying, teaching, and exhortation as separate charismata, gifts. Toward the end of I Corinthians 12, prophets and teachers are listed as different ministries. In Ephesians 4:11, prophets and evangelists are different. Prophets are a different ministry from pastors and teachers in this passage as well.
'Preaching' in the typical New Testament usage of proclaiming the Gospel to those who have not believed it yet, which usually occurred outside of the Christian 'church' is not the same thing as prophesying. Teaching is not the same thing of prophesying.
In the Old Testaments, prophets prophesied and the priests were to teach. The prophet got a fresh message from God, quoting what God said. He might receive dreams, visions, or 'dark sayings.' The priest who taught might expound upon existing prophecies, what Moses prophesied many centuries before, for example.
Prophesying is Revelatory
It is ironic that the Bible never commands us to hear a pastor preach a long sermon and sit quietly by. There is no command that we must sing three hymns and skip the third or fourth stanza. Rather, the one lengthy chapter that tells us what to do in church is about how regular believers in the assembly can take turns using their gifts to edify one another in an orderly manner according to the order Paul describes.
I Corinthians 14
29 Let the prophets speak two or three, and let the other judge.
30 If any thing be revealed to another that sitteth by, let the first hold his peace.
31 For ye may all prophesy one by one, that all may learn, and all may be comforted.
Notice the commandments of the Lord regarding what to do about prophesying indicate that prophecy is revealed. Here, it is revealed on the spot, while the individual is sitting by. He didn't prep it as a sermon.
Content of Prophecies: Predicting the Future?
The 'secular' definition of 'prophesy' is to predict the future. A newspaperman or blogger who wants to wax poetic might call a sports prediction a prophecy. That is not the Biblical definition. Prophesying in the Bible is speaking as the Spirit moves. We often read that the prophet speaks in the name of the LORD, claiming God gave him the message to say. It wasn't a learned message from another prophet, but a message directly from heaven.
Can prophesying be prediction? Clearly yes. There are many examples of this in the Bible. In the New Testament, we see that the prophet Agabus prophesied both a famine that was to come and that Paul would be bound. But prophecies do not have to be predictions. God can say whatever He wants to. If the Spirit of God wants to talk about something coming in the future, He may. If He wants to talk about something in the present, He may. If He wants to say something that other people may not naturally know, He may. If He wants to say something that seems mundane, He may. A prophecy could convey information that we might know if we read the Bible. It might tell us something that we might think should be obvious to us, but that the Spirit wants to say and remind us of, if this is what the Holy Spirit wishes to say.
Revealing the Secrets of One's Heart?
I Corinthians 14
23 If therefore the whole church be come together into one place, and all speak with tongues, and there come in those that are unlearned, or unbelievers, will they not say that ye are mad?
24 But if all prophesy, and there come in one that believeth not, or one unlearned, he is convinced of all, he is judged of all:
25 And thus are the secrets of his heart made manifest; and so falling down on his face he will worship God, and report that God is in you of a truth.
One of the fascinating things about prophesying for those who have experienced it enough, is that some prophecies contain information about the hearer receiving the prophecy that the one giving the prophecy should not naturally know. Just imagine the shock the woman at the well must have felt when Jesus, a stranger, knew about her five husbands.
Can Preaching Be Prophesying, Teaching, Etc.
Caiaphas prophesied, probably without realizing it and without saying 'Thus saith the Lord' or words that revealed directly that he was prophesying. I think there may be occasions where someone is teaching the congregation, 'preaching a sermon' where the Spirit may move him to say something, and he prophesies without saying 'Thus saith the Lord'.
There have been preachers who called out the secrets of individuals in the congregation while they preached. Spurgeon was known to have done such things, even if he may have ironically held to some cessationist leanings.
Covet to Prophesy.
I Corinthians 14 commands believers to covet to prophesy. Paul, writing to the Thessalonians says to quench not the Spirit, despise not prophesyings. Prove all things. Hold fast to that which is good.
It is possible to understand what 'prophesying' means if we really study out the issue in the Bible.