I believe that there are 2 different types of normality.
The first kind is what society agrees on as a whole. It is what most of society has come to accept as what is right or wrong, beautiful or ugly, good or bad, etc.
The other kind of normality is what each of us as an individual has come to accept as right or wrong, beautiful or ugly, good or bad, etc.
The important thing is that in order to function properly in society, individuals need to realize what society's norm is and try to not completely differ from that. I'm not saying to change who you are as an individual. I'm just saying that for instance, murder is (to society) immoral. They look at it as a bad thing. So, if someone goes around committing murders, than they won't be as accepted into society as they might hope to.
When you asked this question as to what normality is, you said how people will say things such as "this isn't normal" or "he or she isn't normal". If two people disagree on what is 'normal', then they are probably talking as what is normal for them as an individual.