Other and another

Other = difference
Another = additional.

Examples in Context:

I would like another glass of water because I drank mine already. I am requesting an additional glass of water.

• Imagine you want to buy a blue towel so the assistant picks up a light blue towel but you don't want the light blue towel. You want the other dark blue towel. (There are choices and you don't want one, you want the other one).

http://www.englishcurrent.com/grammar/other-another-difference/

another + singular nonspecific countable noun

Examples: "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!

other + plural or uncountable nonspecific noun

Examples: "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.

the other + specific noun (singular, plural, countable or uncountable)

Examples: "I have two brothers. One of them lives in Canada. The other brother lives in Japan." "I go to school on Monday and Thursday. I work on the other days of the week." The nouns ("brother" & "days") in both sentences are specific. Re: Example #1: What brother lives in Japan? The other brother. The speaker said he has two brothers. He told you where the first brother lives, so the other brother must be his second brother. This is a specific brother (not general). You know what brother he is talking about. Re: Example #2: What days of the week does he work on? Tuesday, Wednesday, and Friday. The speaker said he goes to school on Monday and Thursday. You know that the other days of the week are Tuesday, Wednesday, and Friday. So the speaker says "the other" because it is clear what other days of the week he works on. Specifically, they are Tuesday, Wednesday and Friday. Hint: Thinking about articles again, the rules for "the other" are the same as the rules for "the" + noun.