Another is used with singular countable nouns to talk about an additional person or thing of the same type as you have already mentioned.
Could I have another cup of coffee?
He opened another shop last month.
It can also be used before numbers to talk about more than one additional thing.
The woman lived for another ten days.
Five officials were sacked and another four arrested.
Other is used with plural nouns, or occasionally with uncountable nouns.
Other people must have thought like this.
They are either asleep or entirely absorbed in play or other activity.
Another / The Other - A Student Page by Yudai Hoshiai