Another-- one more, something different (out of three or more), something different that had not previously been mentioned
"I'd like another orange juice, please."
"Here is another shirt you might be interested in."
Other-- something different or extra; (with "the") not that one (out of two)
"He is some other person." (He is not the one you are talking about)
"I prefer the other one" (not this one, but the other one)
"He has other bicycles in stock". (not just these bicycles)
"I saw him just the other day." (not today, but sometime in the recent past)