USUALLY:
Other means different, extra information, additional, or alternative
Another means one more, an addition (but it can mean different too in some cases.)
I would like another cookie means I would like one more cookie.
We can another day means we can go a different day.
Do you have any other sizes? This shirt is too big. This sentences means do you have different sizes.
Let's go the other way home. There's too much traffic on highway 10. This sentences means let's take a different way home because the traffic on highway 10.
Note: Every other means alternating or every thing but one thing.
For example: Every other day, I go to work. This means Monday, Wednesday, Friday, Sunday, Tuesday....etc One day I go to the work, the next day I don't, the next day I do....
"Every other person I have asked has said no." The speaker has asked 2,3,4,5?? people and everyone said no and now they're telling this to another person while asking them.
This is a hard question to answer because the two words often overlap. Don't get discouraged if it takes awhile to understand it.