Something that's derogatory is insulting or disrespectful. If you make derogatory comments, that means you say things that are unflattering, unkind, or demeaning.
Derogatory means about the same as insulting. Derogatory language is meant to hurt, and it usually does. If you feel offended or insulted by what someone says, the person probably said something derogatory. Racial, sexist, and homophobic slurs are all derogatory. Insults that mean someone is stupid or crazy are derogatory. Making a joke about someone's mother is derogatory.