What does "It doesn't matter" mean if you are in a verbal argument with a person?

That depends on how it's said.  It can be said to be manipulative or the person can just be saying they're giving up or they've decided it's not really that important.

When I say it doesn't matter during our argument it's because the person does not at all pay attention while I'm trying to express myself or concerns regarding our relationship or friendship. Even though it really does matter and it hurts, I say it doesn't matter, never mind. 

mean that it is not important to you because it does not have an effect on you or on a particular situation. you are not concerned at all about its impact, and you are assuring the person that everything is alright.It doesn't matter to you, doesn't mean it doesn't matter to me. From a man: It's really not significant.i ususally say [just f--- it! 

Usually it means

- what we are discussing is not important

or

- what you just said is not important