As Webster 1913 notes, amen is derived from a Hebrew word, and due to its sanctity was incorporated into church liturgy unchanged. Amen is found in both the Old Testament, and the New Testament, being especially common in the book of Deuteronomy, and the works of St. Paul.

The word amen can be used in several different ways, and can vary slightly in meaning. The Hebrew original (transliterated in several different ways) means confirm, or strengthen. Nowadays someone may respond to a saying they particularly believe in with an amen, but the most common use is of course in ending a prayer.

Biblical usage
This practice probably derives from the Lord's Prayer, which Jesus reveals in The Sermon on the Mount, (Matthew 13 v9-13). This, along with texts such as Romans 11 v36 probably led to the early church considering this to be of sufficient importance that the term was imported into all prayers.

While the modern usage of amen is fairly simple, the word is used in several subtly different ways in the Bible. According the the online Catholic Encyclopaedia:1

[I]ts primary use is to indicate that the speaker adopts for his own what has already been said by another[.]
This is of course the meaning of amen that we use today. Another usage, that can be found in the New Testament is of amen simply as an emphasis, placed either before or after the phrase to which it is attached. In some bible translations this is rendered as "verily", or "truly" Perhaps the best example (also demonstrating the doubled usage) is to be found in John 8 v34
Amen, amen, I say unto you, Whosoever committeth sin is the servant of sin.
Another biblical of amen is simply to end an entire book, such as 1 Corinthians. In this case the amen is not applied directly to a previous idea, instead it reaffirms the whole of the gospel.

Liturgical usage
Whereas amen is nowadays used in response to almost all elements of a church service, be they prayer, sermon or creed the specifics were not always the same. Nowadays both the minister and the congregation say the amen, but originally it was uttered only by the congregation, by way of a response. It is thought that the usage was extended after it began to be used by individuals praying on their own.

Even after this (timings, while not precise are within the first few hundred years of the church) some things, especially creeds did not have the amen attached. Since the patristic period usage had become more standardised, although there have been periods in some churches where for example every clause of a prayer was replied to with an amen.

A final interesting note is one of pronunciation. Americans, and those with related accents usually say "Ay-Men", whereas the British say "Ah-Men". I'd be interested in finding out how it's pronounced in other languages, as the original pronunciation, according to one source2 is "aw-mane."


Sources:
  • http://www.gospelcom.net/narramore/concordance.htm
1http://www.newadvent.org/cathen/01407b.htm
  • 2http://www.searchgodsword.org/lex/heb/view.cgi?number=0543
  • http://bible.crosswalk.com/