Another means ‘different one’ or ‘one more of the same kind’. Note that another is one word.

We are going to buy another car. (NOT We are going to buy an other car.)

Another can also mean ‘additional’ or ‘extra’. It is used with singular countable nouns.

Could I have another piece of cake?
Could I get another chance?

Another can be used as a pronoun without a noun. It can also be used with one.

Those apples are really sweet. Could I have another? OR Could I have another one?

Another is not normally used with uncountable and plural nouns. Instead we use more.

Compare:

Would you like some more meat? (NOT Would you like another meat.)
Would you like another cup of coffee? (More natural than ‘would you like more cups of coffee?’)

Another, however, can be used before a plural noun in expressions with few or a number.

We will be here for another few weeks.
We will need another five chairs.

Other

Other refers to the person or thing not the same as that already referred to.

Give this to Jack and that to the other boy.

Other has a plural form.

You two boys can go home but all the others must stay here.

Note that other does not have a plural form when it is used with a noun.

Where are the other books? (NOT Where are the others books?)

Other is not used an adjective to mean ‘different’.

I would prefer a different color. (NOT I would prefer other color.)