The quote is incorrectly formed. In written text it would be acceptable with different punctuation. When spoken, I would consider it factually incorrect. In writing:
You can use 'arithmetic' to refer to the process of doing a particular calculation.
You can use the word 'arithmetic' to refer to the process of doing a particular calculation.
Of course, when spoken, there are no quotation marks, which is why you need to add "the word" to make your meaning clear.
In any case, in this usage to refer to means to describe or denote. That definition is not in some of the online dictionaries I looked at, but it is in my offline dictionary and it is the best definition I found.
Hopefully someone more versed than I am in naming verb tenses can explain why a word means something but you use a word to refer to something.
As for other uses of the word:
I refer you to this page: ldoceonline.com/dictionary/refer - Manoochehr
is a great example and a good link to more examples of other meanings.