I think "another" refers to an additional thing whereas "other" refers to a different thing. "The other" is equal to "other". ex. "Another day, another dollar." "I met up with Marcus the other day." "Sometimes life is good, other times, not so much." "Can I have another glass of wine?" "Jack stayed inside, while the others went out to play." ^^^ In that last example you can also see that "other" can refer to people whereas "another" can not.