september
The word 'September' is a direct loanword from English to German. It is used in the same way to refer to the ninth month of the year in the Gregorian calendar. In German, nouns are capitalized, so 'September' is written with a capital 'S'.
I moved to Berlin in September.
In this phrase, 'im September' translates to 'in September', denoting the time period when the speaker moved to Berlin.
My birthday is on September 12th.
The phrase 'am 12. September' means 'on September 12th'. This is how dates are typically expressed in German, with the day number preceding the month.
School starts in September.
Here, 'im September' translates to 'in September'. This refers to the commonly understood concept that the academic year begins in the month of September.