The word mad has several meanings:

1. Angry or annoyed. When your friend breaks a promise, you might feel mad at them. If your little brother scribbles in your favorite book, you'd probably get mad. This is the most common meaning in everyday American English. British English speakers more often use “mad” to mean crazy rather than angry.
2. Mentally ill or irrational. In older stories, you might read about a mad king who made bizarre decisions that endangered his kingdom. Scientists sometimes use names like mad cow disease because the illness can cause animals to behave erratically.
3. Wildly enthusiastic or extremely eager. When you're mad about soccer, you think about it constantly and play whenever you can. A collector might be mad for vintage stamps, filling albums and hunting for rare finds.
4. Frantic or frenzied. During a mad dash to catch the school bus, you're running wildly and desperately. A crowd might make a mad scramble for free concert tickets.

The phrase like mad means doing something with intense energy or speed: working like mad to finish homework, or pedaling like mad to win a bike race.