Imagine a plate of different kinds of cookies. Your friend offers you one. You respond:
1a. Thanks! May I have another?
1b. Thanks! May I have another cookie?
2a. Thanks, but I'd prefer the other one/the other kind. (You are telling him you don't want the particular kind of cookie he offered.)
2b. Thanks, but I like (the) other cookies better.
3. Thanks, and I'm so hungry I'll eat all the others!
Another is usually singular.* It may be used by itself or with a singular noun:
I'll have another/I'll have another cookie.
Other can be singular or plural. It is preceded by "the" and is generally followed by a noun.
I like the other cookie/cookies.
I like the other one/ones.
Others is plural. It is not followed by a noun.
I like the others.
*Another can be used with a plural noun if it's preceded by a number, or certain fixed phrases:
I'll want a cookie in another two hours/ couple of hours.