Learn Italian: Lei è un'amica italiana. - She is an Italian friend.

un' is short for una. When a feminine word starts with a vowel, like amica, the una drops its -a and gets an apostrophe so the two vowel sounds don't crash into each other.

Would we still use an apostrophe if the friend was male?

No, a male friend is un amico. The masculine word for "a" or "an" is just un, and it never takes an apostrophe. The apostrophe only happens when the feminine una gets shortened.

Why does italiana come after the word for friend?

In English we put the adjective first ("Italian friend"), but in Italian, descriptive adjectives usually come after the noun they describe. Notice how it also ends in -a to match the feminine noun amica.

Could Lei mean the formal "You" in this sentence?

Yes! Because it's at the start of the sentence, it has to be capitalized anyway, so we can't tell if it means "She" or the formal "You". Depending on the context, this could also mean "You are an Italian friend."