What does it mean if you're half a nationality for example half-Irish or half-Italian?

If you refer to a person of dual ethnicity, this is a person who likely has a parent from one country and another parent from a different country. In the case given, perhaps the father was from Ireland and the mother was from Italy or they have heritage from these places.

The phrase "half a nationality" makes no sense. Nationality refers to your official citizenship documents. You are either a national of X country or you are not. A person cannot be a "half US-citizen"; he either is a US citizen or he is not a US citizen. Now, a person can have multiple nationalities, but he has each of those nationalities in full. For example a person can have French nationality and Dutch nationality, but he is not half of a French national and half of a Dutch national, he is both.

If one of your parents is from, say, Italy and the other is from Ireland, then you are of Italian and Irish descent. It can be said that you are half-Italian and half-Irish. Some people will call themselves part-this and part-that even if both parents were born in the United States.