The Greek verb kataphroneo means to think down upon, to despise, to treat with contempt, or to disregard entirely. The prefix kata- (down) combined with phroneo (to think) creates the image of looking down one's mind at something.
Jesus uses kataphroneo to describe the impossibility of serving two masters: no one can serve both God and money — they will inevitably despise one (Matthew 6:24). Paul warns Timothy not to let anyone despise his youth (1 Timothy 4:12). Hebrews 12:2 says Jesus despised the shame of the cross — meaning he regarded it as beneath his dignity in comparison to the joy set before him.
Theologically, kataphroneo warns against contempt — toward God's discipline (Hebrews 12:5), toward weak members of the church (Matthew 18:10), or toward spiritual things in favor of material ones.