Generally you use 'other' when making choices or comparisons.
"Do you like this picture?" "No, I like the other"
- "Are you single/ married / divorced / other?"
"That is my friend and the other person is my brother."
"The other show on television is good, but this one is bad."
'Another' is used for additional things.
"I like this beer, I want 'another'"
"I want to travel to 'another' country (I am already in a country but I want to see MORE)
"I want to learn 'another' language (I already know one I want to learn MORE)"
Here are two sentences:
"I want to watch another movie."
"I want to watch the other movie."
The first sentence implies that I have already watched one or more movies but now I want to watch an additional movie. The second implies I don't want to watch this movie, I'd prefer to watch a different one.
Hopefully this helps a bit!