I'm not sure I can explain it well, but I'll give you a couple of dialogue examples that might help:
OTHER:
*there are two books on the table*
person 1 "could you hand me that book?"
person 2 "the red one?"
1 " no, the other one"
^^^ normally 'other' is used like this: when there's two choices, one is considered 'the other'
---- here's another way to use it
1 "other than ice cream, did you want anything from the store?"
2 "yes, there's another thing- please get some milk"
^^^in this case, 'other' means 'besides'
---- one more example
1 "have you seen John lately?"
2. "yes, I saw him just the other day"
^^^ i don't know how to explain this one- but we use it to talk about a recent day
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ANOTHER: another normally just means 'one more',
1. "Do you want another cookie?"
2. "Sure, I'll have another"
unlike saying 'one more', it doesn't specify *just one*, you could have another, and then another after that
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