I go to a Gemara shiur, and the rabbi always says "the first aman deamar", "the second aman deamar"... What does it mean exactly? What is the meaning of the different words?
The words you're actually hearing are "man de'amar", or מאן דאמר in Aramaic.
"Man" here actually has the same meaning in Aramaic as in English - man. "D'amar" is like the Hebrew שאמר. Translated, it means "who says".
The whole phrase together means "the [first/second] one who says".
It's often used as a noun when talking about different opinions - "this man d'amar says x, while the other man d'amar says y". Or: "according to the first man d'amar...".
While not the case you're talking about, aman d'amar (אמאן דאמר) is also an Aramaic phrase. The initial א means "on", so it means "on the one who says". Maybe i'm not listening to the right shiruim, but i haven't heard this used so much in English.