Amen is a Jewish word that enjoys a level of international fame comparable to shalom or bagel. It signifies emphatic assent or agreement to something just said - as in "Amen to that!"
Amen is Hebrew for "firm," "straight" or "true." It comes from the verb "aman" - to "support," "confirm" or "approve." When we say amen, we confirm or support the truth of whatever we are responding to.
We say amen virtually any time when we hear another Jew saying a blessing (but not to our own berachah, or to that of a child who is learning to say blessings, or when the berachah is unnecessary, or when we haven't actually heard the berachah.)
Saying Amen, one should consciously assent to the content of the berachah. It's important to take care not to say it too hastily so that we drop a sound, eg "'men," or cut off at the end, "ame-" (Talmud Berachot 47a).
Amen is related to the word "emunah," which means "faith" or "trust," and also to the word for nursemaid, "omenet." As the Chief Rabbi likes to point out, whereas the word "faith" evokes a mental state of belief, the Hebrew emunah denotes a relationship of mutual faithfulness and reliability. Faith suggests a state of mind which you either have or do not, whereas emunah is a relationship that is acquired and strengthened through commitment and practice.