Another means “one more noun” whereas “an other” means “a different noun”.
We need another tennis racket. One is not enough.
There is an other type of tea that I like. What is it called?
Another is much more common in English than “an other”. It means one more or additional. We use “other” when we have more than “one more”.
We have another game on Saturday. (one more)
We have other games later on in the season. (more than one more)
We use “an other” when there is something else other than the noun we are talking about that is different.
No, No, not this one, There is an other one that is better. (a different one from the one we are talking about)
We use “an other” to show that there is one more(but it is one more out of a group of many) that could be suitable.
A.N. Other can be used as a placeholder when we don’t know the name of the person.
I have seen this mainly on match day sports programmes where the name of the player is unknown at the time of printing.