Other = difference
Another = additional.
Examples in Context:
I would like another glass of water because I drank mine already. I am requesting an additional glass of water.
Imagine you want to buy a blue towel so the assistant picks up a light blue towel but you don't want the light blue towel. You want the other dark blue towel. (There are choices and you don't want one, you want the other one).
Imagine you are eating a piece of cake at a new restaurant. It isn't very good so you say, "The cake at the other restaurant is better than here".
You want to buy fish and you are comparing prices at the market. The first kiosk is charging 4Euro a kilo but you want to see more. So you could say, "Let's go to another kiosk to compare the prices".
At the second kiosk they are charging 6Euro a kilo. So then you could say, "The fish is cheaper at the other kiosk".
An exception to this rule
When you use "another day" and "other day" the meaning changes.
The other day I went to Portugal.
This means that previously (in a non specific way) you went to Portugal.
We can go to Portugal another day.
This means we can't go today but we can go another day (we can go a different day).
This is probably why some people get these two words confused. Just remember that generally speaking;
another = additional
other = different
Except for when you are referring to a day...
another day = a different and separate day usually in the future.
other day = a different day in the past.
Cheers,
Niamat
