other + plural or uncountable nonspecific noun
"Other people have problems, too." [people = plural noun]
"This book has other information." [information = uncountable noun]
The nouns in both sentences are not specific, just like with "another". The speaker doesn't specify which other people have problems, or what other information the book has. The rules are the same as "another" except that "other" is used before plural or uncountable nouns.
Hint: If we think about articles (a/an/the) again, then remember that we use "other" before a noun that would NOT need an article.
another + singular nonspecific countable noun
"Let's meet another day."
"I'd like another piece of cake."
The nouns ("day" and "piece") in both sentences are countable and singular (e.g. not with an 's') nouns.
The nouns in both sentences are also not specific. This means that the speaker doesn't care which day or piece of cake he gets; he just wants a different one. He wants another one, but he hasn't said (or it isn't clear or important) which one.
Hint: If you understand English articles ("a/an/the"), then think of "another" as "an + other." You can use "another" before a noun whenever you can use "a(n)" before a noun. The rules are the same. Another = an other!