What does “We use the formal lei” mean? [closed]

There was the following passage in an article written by Jhumpa Lahiliri, 2000 Pulitzer Prize winner that came under the title, “Teach yourself Italian” in The New Yorker (December 7, 2015):

“I’m very fond of my teacher. Although for four years we use the formal lei, we have a close, informal relationship. We sit on a wooden bench at a small table in the kitchen. I see the books on her shelves, the photographs of her grandchildren.”

As I was unable to catch the meaning of “We use the formal lei,” I consulted English dictionaries online, and found that almost of all dictionaries provide nothing more than a definition of “lei” as “a necklace of flowers that is given to a visitor in Hawaii” like Merriam-Webster.com/dictionary does.

But a Hawaiian necklace doesn’t seem to have anything to do with learning Italian or fit well in the line, “Although for four years we use the formal lei” to me. What is “lei” here? Why is it shown in italic? Why is it the formal lei even they have an informal relationship?

It's not English. It's Italian. It's the formal version of "you" in Italian (here).

The italics indicates that the word is being used metalinguistically, that is, that the word is being mentioned rather than used.

The author is saying that he and his teacher used the formal version of "you" even though they had an informal relationship.