1. 'Another' is just the two words 'an' (indefinite article) - meaning 'one' -and 'other' (adjective) written together as one word.
An (one) + other = another
Do you know when to use the article 'a' or 'an'? Of course you do - that's easy. You use the indefinite article with singular countable indefinite nouns: an apple, a book, and so on. Well, using the adjective 'another' is exactly the same.
An apple ---> 'an other apple' ----> another apple
A book ----> 'an other book' ---> another book.
2. You use 'other' on its own with all other nouns - plural nouns ('books') and uncountable nouns (for example, 'information'). With these words, there is no article, so we just say 'other books' and 'other information'. We also use 'other' on its own after other determiners, for example 'My other book' or 'That other apple'.
I hope that makes sense. I would be a multi-millionaire if I had a penny every time someone has asked that question.
And the answer is so simple: an + other = another.