Imagine that in Spanish instead of using the word "otro" you had to use the word "diferente". Would you use "un" / "una"? Then it's probably "another" in English.
Relying a bit on Google Translate:
Ayer vi a otro amigo.
Ayer vi a un amigo diferente = I saw a different friend yesterday
Therefore: I saw another friend yesterday.
El mago puede usar magia de fuego y también otra magia.
El mago puede usar magia de fuego y magia diferente también = The wizard can use fire magic, and different magic too
Therefore: The wizard can use fire magic, and other magic too.
In other words, it is a complexity of Spanish and Portuguese grammar which often makes you drop the indefinite article with the word for "other", whereas with a different adjective you would include the indefinite article. "An other" normally becomes "another" (possible exceptions include when "other" is used with special emphasis as a noun).
"Other" can also be used in the phrase "other than" (to mean "aparte de").