"January is a month of the year."
Translation:Janvier est un mois de l'année.
It all has to do with how you view the year. Is it being used for counting or stating a date (an) or is it being used to focus on events that take place within in (année)?
Here is a link that may help.
The word an tends to view the year as a single, objective "point" in time for the purpose of counting, stating a date etc;
The word année is more subjective: it is used to describe a particular year, or focus on the elapsed time and events that occur in a particular year or years.
For months of the year and days of the week, you can use month/day + est.
Juin est le meilleur mois pour un mariage. → June is the best month for a wedding.
For words like aujourd'hui, hier, demain, use c'est. These words are not nouns in French, but adverbs, so you need "c'est."
Aujourd'hui, c'est mon anniversaire. → Today is my birthday.