Other... Another... Others... These words are very confusing for students of English, let's explain the difference, it's not that difficult. First we need to make a few points clear:

ANOTHER = An + Other So you must apply the rules of usage for the indefinite article A/AN. 

OTHERS is a plural form Remember that only nouns have plural in English, adjectives/determiners have no plural so you can't use OTHERS when it goes in front of a noun. You can only use if it is a plural pronoun.

OTHER ONE/S or SOME OTHER  If we use it as a pronoun, we can use just OTHER/S, but we can also add the pronoun ONE/S for countables and SOME for uncountables: The others are broken = The other ones are broken / Not this wine, I'll try other = I'll try some other.

Careful with the pronunciation: OTHER /ʌðə*/  ANOTHER /ənʌðə*/

CHOOSING THE RIGHT ARTICLE

The most important thing is to choose the right article. Think of the sentence without OTHER; what article would you use? then use that same article with OTHER:

- I want a book  I want another book
- Can I have a beer?  Can I have another beer?
- This wine is not sherry, it's Italian  This other wine is not serry, it's Italian.
- I saw animals  I saw other animals
- Where are the boys?  Where are the other boys?
- I've got (some) CDs  I've got (some) other CDs
- Do you need (any) chairs?  Do you need (any) other chairs?
- I have read no books  I have read no other books.

CHOOSING OTHER or OTHERS

When OTHER goes with a noun it is never plural because it is an adjetive. But when it doesn't go with a noun, then it is a pronoun, and it may be singular or plural like any other noun.

Remember also that when it is a pronoun we can add ONE (and we certainly prefer that for the singular), but in that case ONE takes the plural and OTHER behaves like an adjective, so it is never plural (OTHERS ONE - OTHERS ONES - OTHER ONES).

We can't use the pronoun ONE for uncountable nouns, so in that case we use SOME (and we also prefer SOME OTHER).

Adjetive (other)
- I need another pencil
- Do you have other books?
- These are the other friends I told you about.
Pronoun (other/others):
- This glass is broken, I need another. = This glass is broken, I need another one. (better)
- This wine is terrible, bring me other. = This wine is terrible, bring me some other. (better)
- Here are 5 chairs. Do you need any others?  = Here are 5 chairs. Do you need any other ones?
- Put some boxes here and the others in the garage. = Put some boxes here and the other ones in the garage.

Note: for the countable singular pronoun it is possible to use "other" (I have 2 chairs but I need 3. Give me other), but that is not common and complicates the rules we have here, so you'd better forget about it and use "another" or, better still, "another one".

ADVANCED

Everything said is good enough for normal uses, but if you already know that, and have no problems using OTHER, maybe you want to learn about a few special cases.

DIFFERENT or EXTRA

When we refer to a plural or uncountable noun, we tend to use OTHER/S if we mean "different" and we tend to use MORE if we mean "extra, additional".

Compare:

- I don't like these books, I want to read others     (= different books)
- These shoes are too ugly. Can I see any others?    (= different shoes)

- This coffee is delicious, can I have more?      (= extra coffee)
- I bought some books but I want to buy more.    (= extra books)

But for countable singular we can use ANOTHER in both situations.

- This hat is too big for me. Can I try another one?    (= a different hat)
- Mmm, I love these cakes. Can I have another one?    (= an extra cake)   Also: Can I have more?

PEOPLE

We often use (THE) OTHERS meaning "(The) other people"

- It's only you? I was expecting five, where are the others?
- I like big cars, but others prefer small cars.

OTHER PEOPLE often means "people besides oneself"

- Don't be selfish. Why don't you think more about other people?
- Tom is a great person. He's always helping other people.

NUMBERS

We can use OTHER before or after the numbers. Before the noun it has no plural form because it's an adjective (other two), but after the noun it is always plural because it's a noun (two others), except when it goes with a noun, in which case it is again an adjetive (two other boxes).

- I have 6 boxes, two here and four others at home. The other four boxes are bigger. But I think I need two other boxes to pack all my books.

- We have 10 boxes but I can only see 7. Where are the other 3? (or: Where are the three others?)

TWO OTHERS or THE OTHER TWO?

But many people make a little distinction between both positions: "Two others" (or three others, etc.) means "two more", an additional number. "the other two" or "other two" means "the rest of them".

- If you buy this pack of five bottles, I'll give you two others for free / two other bottles for free, so you'll have 7 bottles for the price of five! (you buy 5 and get 2 additional bottles) [in "two others", OTHERS is apronoun, in "two other bottles", OTHER is an adjective, so it has no plural]
- I have 5 children, 3 are here and the other 2 are at home with my mother / the other two children.... (5 = 3 + the rest, 2)

But if you find this confusing don't worry, some people make no distinction, so they think "the other 2 books" and "the 2 other books" are the same.

NOTE FOR SPANISH SPEAKERS

Lo más diferente para un hispanohablante es en el caso del artículo indefinido (un/a, unos/as). Ahí el inglés, en el singular, hace algo peculiar, escribe AN y OTHER junto (ANOTHER). El español también hace algo peculiar, elimina el artículo, tanto si es singular como plural:

- Quiero un libro Quiero otro libro  (No: *Quiero un otro libro)
- Dame una moneda Dame otra moneda (No: *Dame una otra moneda)
- Toma unos caramelos Toma otros caramelos (No: *unos otros caramelos)

Sin embargo con el resto de determinantes (artículos, etc.) hacemos lo mismo que en inglés, seguimos usando el mismo artículo que había (o a veces tenemos que cambiarlo por otra cosa):

- Dame el libro Dame el otro libro
- Cierra las ventanas Cierra las otras ventanas
- Dame algo de vino Dame otro poco de vino
- Necesito algunas sillas Necesito algunas otras sillas
- No conozco a ningún niño aquí No conozco a ningún otro niño aquí

Lo mismo vale para el pronombre. En todos estos ejemplos si eliminamos el sustantivo nos quedamos con OTRO en función de pronombre (quiero otro, dame otra, toma otros, dame el otro, cierra las otras, dame otro poco, necesito algunas otras, no conozco a ningún otro).