1. First, it might be helpful if you remember that the word "another" consists of two words:
"an" = one + other. Thus, "another" means "one other."
2. I hope that the following dialogues help you:
Teacher: I have two pencils. One is yellow. One is red. I'm going to give you the yellow one.
Student: I'm sorry, Teacher. But may I please have the OTHER one. (That is, there are two things.
I do NOT want the thing that you gave me. I want the thing that you did NOT give me.)
Teacher: What's the problem, Tommy?
Tommy: This book is full of marks on the pages. May I please have ANOTHER book? (That is, I
have something. But this something does not satisfy me. I want to exchange it for something
that is similar but different.)
I know that these words can be confusing. I have made up some other examples:
3. The right shoe fits comfortably, but the OTHER one doesn't. ("Other one" = left shoe.)
4. This doughnut is stale. May I please have ANOTHER one. (One other donut -- not this one.)
5. You have offered me five hats, I do not like any of them. Do you have any others? (THE other = two things; any otherS = three or more things.)