'Another' is simply the words 'an' and 'other' written as one word.
The answer isn't quite as simple as a distinction between singular and plural, however. Basically, you use 'another' in any situation when you would use 'a' or 'an' before the noun. You use the indefinite article 'a' or 'an' for singular countable indefinite nouns, and you use the word 'another' in exactly the same way.
Singular countable indefinite ----> a book ------> ANOTHER BOOK
Singular uncountable indefinite ---> information ------> other information
Plural countable indefinite: ------> books ---------> other books
Singular countable definite -----> the book -------> other book
Singular uncountable definite ----> the information -----> other information
Plural countable definite ----> the books --------> the other books
As you can see, 'other' is an adjective which you can put before any noun. If there is a determiner (a, an, the, this, those, that, these, my, your, his etc) before the noun, the word 'other' goes between the determiner and the noun e.g.:
my other book, this other book, those other books, that other information
But, instead of saying ' a other book' or 'an other book', we write the two words 'an' and 'other' as one word ---> 'another book'. It really is that simple.