The word other is an adjective. It refers to something different.
The teacher held a textbook in one hand and a pencil in the other hand.
The word "other" is often used with "the." It can be used with singular or plural nouns:
We crossed to the other side of the street.
I liked the lesson, but the other students thought it was boring. the others students
Let's finish this task and leave the other tasks for tomorrow. the others tasks
thers is a noun, and we use it when we DON'T say the noun specifically. It can only be used for plural things:
I liked the lesson, but the others thought it was boring.
Let's finish this task and leave the others for tomorrow. or the other tasks (implied)
The word another is used when you are talking about one additional thing:
I ate two slices of pizza and I'm still hungry, so I'm going to eat another one. (one more slice of pizza)
The Harrisons just had another baby - now they have five children!
(one more baby)
It can also be used to describe something different, like a/an + other:
I don't understand. Could you give me another example?
So what's the difference between "the other" (when use for a singular noun) and "another" (also singular)? "The other" is specific - it is when there are only two options. "Another" is general - it describes one of many options.
There was a red shirt and a blue shirt. I look terrible in red, so I bought the other shirt. (= the blue one, the only option different from the red shirt).
I don't like this red shirt. Do you have it in another color?
(another color = one other color among many possibilities)