The hebrew word, "Amen", is made up of three hebrew letters - Aleph, Mem, Nun. This is sometimes said to stand for "El Melech Ne'eman" which means "G-D is the faithful King". In modern / Sephardi pronounciation, it's pronounced "Ah-men". In Ashkenazi pronounciation, it's often pronounced "O-mayn".
In saying "Amen" to a prayer or a bracha, it is as if the person is saying "Yes, I agree with what the prayer or blessing says". So, for example, the first few words of the Kaddish prayer say (roughly) "Great and Holy is his is (G-D's) name", to which the congregation respond "Amen".
In Synagogue, therefore, it's usual for everyone present to respond "Amen" at the end of every bracha. Indeed, there are certain parts of the prayers (such as the public recitation of the Amidah prayer) where it's essential for at least a minyan of men to respond Amen to each of the blessings in the prayer for it to be "valid". So the story goes, the Great Synagogue of Alexandria was so big that not everybody could hear the person leading the prayers, so someone would raise a flag at the points that everybody needed to say "Amen". Although whether it's valid to say "Amen" to a bracha you haven't actually heard (or, to take a more modern example, through a PA system) is a separate matter for discussion!
Furthermore, in some cases, it's considered that responding Amen to a bracha is as good as saying it yourself. For example, when making Kiddush at home on Shabbat, one person usually recites the prayer, ending with the bracha over the wine, and everybody present says "Amen". Everybody can then drink the wine, without needing to make the bracha themselves, as they have accepted what the bracha says by saying "Amen".
Amen has also been used as a response to a curse - for example, in the Torah, Deuteronomy 27:16 "Cursed is anyone who dishonours their father or mother - and all the people shall say, Amen". In this case, the people are accepting the outcome of the sin.
In general one does not say Amen to a bracha that you make yourself. One well known exception is the fourth blessing in Birchat HaMazon, the full grace after meals. However, firstly, this isn't a blessing in the "conventional" sense of the word, and secondly, one traditionally pauses briefly between the end of the sentence and the word "Amen".