Think of another as an + other. We use this word in the same situations where we would use the indefinite article (a/an), like with undefined singular countable nouns. We use 'other' in situations where we wouldn't use an article, such as with general plural or uncountable nouns, and 'the other' in situations where we would use the definite article.
"Would you like another cup of coffee? I have other milk if you don't like cow's milk."
"Thanks, another cup of coffee sounds great. Is this my slice of cake?"
"No, yours is the other one, this one is mine."