Another, other, the other, others, and the others all come from the same root, and choosing between them depends on two questions: is the item singular or plural, and is it one specific remaining item or just any additional one? Learners who never sort this out keep reaching for "other" as a catch-all.
All of these words come from other, and two questions decide which one you need: singular or plural, and one particular remaining item or just an extra one?
Another is an + other, so it points at one more item, or one more measured amount, from an open set (Can you play another song? / We waited another ten minutes), and it never takes the. Other is the adjective form and never adds -s in front of a noun, whether that noun is singular or plural (my other glove, the other students). Drop the noun and the word becomes a pronoun: the other is the one specific item left (I found one shoe — where is the other?), others means some more from an open set (Some people study at night, others in the morning), and the others is all the rest of a known group (One chair is broken, but the others are fine).
Use another for one more or a different one when the set is open and you are not pointing at a particular item
He finished one task and immediately started another.
Use the other for the one specific item left, which is why it takes the
One of my shoes is under the table. Where is the other?
Use the other in front of a plural noun, or a number, when it covers all the remaining members of a known group
One student stayed behind and the other students went home.
Use other as a plain adjective in front of a noun, with no -s on it
Show me some other photos.
Use others on its own for some more people or things from an open set
Some people believe in luck, while others do not.
Use the others on its own for all the rest of a known group
One student is absent today, but all the others are present.
Practice
You took one cookie. Why not take ?
I will take this bag, but I do not need the one.
I only found one shoe. Where is the ?
He has two cars. One is new, and is very old.
I was given two job offers. One came with a higher salary, and came with better hours.
I called two friends yesterday. One answered, but did not.
She looked at three dresses. One was too big, was too small, and one was perfect.
He finished one task and immediately started .
Some children are shy, but are very talkative.
Out of the three boys, one is tall, and two are short.