It’s a mad world means that it’s a crazy or absurd world.
Indeed, mad is an adjective that has several different meanings.
It can mean to be mentally ill:
The artist went mad in his old age.
or very foolish:
You must be mad to turn down such a good job.
Or very enthusiastic about something or someone:
He’s mad about football.
Or, like the girl in the photo, to be angry:
She was so mad with her sister that she refused to speak to her.