Deranged means insane. You've probably read headlines about deranged murderers, but someone might also be called deranged for bizarre behavior like wearing a bathing suit while skiing.
Crazed, insane, demented, unbalanced, and unhinged are all synonyms for deranged. The literal meaning for deranged was originally “out of order," so think of it as describing someone whose thoughts are not in order. The madwoman in Bronte’s Jane Eyre is a classic deranged character: “In the deep shade, at the farther end of the room, a figure ran backwards and forwards. What it was, whether beast or human being, one could not, at first sight tell.”