Often, we use the word "mad" not to mean mentally ill, but when we think something (or someone) is a little unbalanced. We also use it to comment on things that we don't understand or which seem illogical to us.
"Mad" is often used informally and in spoken English.
barking (barking mad)
Dogs bark at the moon, so if you say someone is barking, or barking mad, you think they're acting illogically. We often use this expression affectionately.
"My aunt is barking. She wants to sail around the world in a yacht."
be / go stark raving mad
This means to go completely crazy. You need to use all three words in this expression "stark raving mad".
"Did you hear what Sue did? She gave away everything she owned to live a more "balanced life", she said. I think she's gone stark raving mad."
be going mad
If you think you're going mad, you doubt your own sanity or reason
"I think I'm going mad. I'm sure I left my book here, but I can't find it at all."
drive someone mad / drive someone up the wall
If you drive someone mad, your beahviour is irrational to another person and it is frustrating or annoying them.
"He keeps changing his mind. It's driving me mad!"
be hopping mad
This means to be very angry.
"Jim's hopping mad. He got his third speeding ticket in a week!"
be in a mad rush
This means to be very busy.
"I can't help you right now. I'm in a mad rush this morning, trying to sort out all the paperwork before our trip."
do something like mad / like crazy
If you do something like mad, you do it a lot, or with a lot of enthusiasm.
"I've been working like mad recently."
"They've been studying like mad these last two weeks."
mad scientist (also mad professor)
In popular culture, a mad scientist spends all their time doing strange experiments.
mad about someone / something
This is when you love someone / something.
"She's mad about this new boy in her class."
mad at someone
Be careful with the prepositions after "mad". If you're mad at someone, then you're angry with them - not in love with them.
"Are you mad at me for breaking your favourite mug?"