Learn Farsi: اردیبهشت ماهِ دوم است.

Just like regular adjectives, ordinal numbers like دوم (dovvom, meaning second) follow the noun they describe and connect to it using the ezāfe (the short 'e' sound). So 'the second month' becomes ماهِ دوم (māh-e dovvom).

Could I say دومین ماه (dovvomin māh) instead of ماهِ دوم (māh-e dovvom)?

Yes! In Farsi, you can form ordinal numbers in two ways. You can use دوم (dovvom) after the noun with an ezāfe, or you can add the suffix -ین (-in) to make دومین (dovvomin) and put it before the noun without an ezāfe. Both phrases mean exactly the same thing.

Is اردیبهشت (Ordibehesht) just the Farsi translation for February?

No, اردیبهشت is the second month of the Solar Hijri calendar, which is the official calendar of Iran. The Persian year starts on the first day of spring (Nowruz), so this second month corresponds roughly to mid-April through mid-May.

Does the word ماه (māh) have any other common meanings?

Yes, besides month, ماه also means moon. Because early calendars were based on lunar cycles, the word for moon became the word for month, very similar to how 'moon' and 'month' are related in English! You might also hear it used as a compliment to describe someone as beautiful or sweet.