This is the 18493rd most frequent Russian word.
январские
"Январские" means "January's" or "related to January."
In this sentence, 'январские' is used as an adjective to describe the type of frosts ('морозы'), linking them specifically to the month of January.
Январские морозы часто бывают очень сильными.
January frosts are often very severe.
Here, 'январским' functions as an adjective modifying 'праздникам', indicating that the holidays referred to are those occurring in January.
Мы готовимся к январским праздникам заранее.
We prepare for January holidays in advance.
In this example, 'январские' qualifies 'дни' as being in January, emphasizing a temporal characteristic of the days.