This question is too broad to be dealt with in detail here so I'll just give a couple of pointers. other , when used without "the", is always used before a plural noun. (other people, other things etc) other is sometimes used before a singular noun but then it requires 'the'.
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Ardyhow to distinguish :You cannot understand the differences among these until you understand the meanings and uses of the, a, an, any, and some. The word other has little to do with it. You can substitute different and have practically the same meanings, and
-ANOTHER
-OTHER
-ANY OTHER
-SOME OTHER
-THE OTHER