These words refer to something different, remaining, or additional and are placed before the noun.
Another is used with singular nouns, other with singular or plural.
-There are other jobs you could try.
-Where's the other packet of cereals?
-Is there any other bread?
-Have another cup of tea.
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Another is used when referring to "one more".
-Please pass me another napkin.
-Another soldier has died as the result of enemy fire.
"The other" is used to differentiate two or more objects.
Suppose I am carrying a bowling ball, a tennis ball, and a soccer ball. My friend, who is somehow unfamiliar with all of these objects, asks, "What is that ball used for?" I raise the tennis ball, asking,
"This one?" If this is not the one he is referring to, he could say, "No, the other one." Suppose I guess incorrectly again. He could then say, "The other one; the small green one."
"Other" is used in the same way as "the other."
For example, "Other than that (something horrible), my vacation was wonderful."
Or, "My other cousin is a jerk; I like this cousin more."