I'm writing this in order to increase our wiki.
Cambridge Dictionary says: to be quick to understand and react to things
UsingEnglish says: If someone's on the ball, they are well-informed and know what's going on in their area of responsibility or interest.
Phrase Finder says: To be alert; in command of one's senses.
- This guy is really on the ball.
- I didn't sleep well last night and I'm not really on the ball today.
- Both John and Mary have a lot on the ball. They should go far.
- I think I'd do better in school if I had more on the ball. I learn slowly.
- If you were on the ball, this wouldn't have happened.