"You want a piece of me?" is a very confrontational thing to say. They're basically asking if another person (or people) want to fight them. It tends to be said by a very confident person who wants the other person to know they're more than willing to solve things violently. It can also be used in a more casual sense and in a nonviolent manner, but this is very dependent on context. Like if a family member challenges you to a game you're very good at, it could be used in that manner as well. There are also other uses for the term. This is often used in an emotional sense, such as "A part of me will always love ..." If someone says that, it implies that they have moved on from whatever or whoever they refer to, but still care about it.