It's really the difference between the definite article (the) and the indefinite article (a or an). For example, two men are in a shop. One man is called John, but you don't know the name of the other man.. so you ask John "What is the name of the other man?" You are talking about a specific person. Second example, you stop a man in the street and ask him the way to Buckingham Palace. He doesn't know the way. So he says to you: "I'm afraid that I don't know, you will have to find another person to ask." He is referring to somebody else but not to a specific person. So, to talk about a specific person or thing = the other. To talk about a non-specific somebody or something else = another.

Another = an + other = one other So, you use "another" with singular things, and never with plurals. Also... "say me" is not possible in English. The phrase you want is "tell me..."

other- one or some of the rest, another- a specific one of the whole.