Learn Farsi: آذر ماهِ پاییز است.

That is the Ezafe particle! It links a noun to its modifier. In ماهِ پاییز (mah-e payiz), it literally translates to 'month of autumn,' which we say as 'an autumn month' in English. The Ezafe is usually a hidden vowel sound in the Persian script, though sometimes a small symbol is added for beginners.

What exactly is آذر (Azar)? Does it translate to November?

آذر (Azar) is the ninth month of the Solar Hijri calendar, the official calendar in Iran. It spans from late November to late December. Because the Iranian calendar doesn't line up exactly with the Gregorian one, we don't translate it to an English month; we just use the name آذر.

The English translation says 'an autumn month,' but there is no word for 'a' or 'an' like یک (yek) in the Farsi sentence. Is it missing?

It is not missing! In Farsi, when you are describing what category something belongs to, you often don't need the indefinite article یک (yek) or the suffix ای (-i). Saying آذر ماهِ پاییز است is perfectly natural for expressing 'Azar is an autumn month.'

How would this sentence change in everyday, spoken Farsi?

In casual spoken Farsi, the formal verb است (ast) meaning 'is' is almost always shortened to an 'e' sound attached to the previous word. So instead of 'Azar mah-e payiz ast', you would hear 'Azar mah-e payize'.