1. Careful to avoid embarrassing or upsetting others.
2. Tactful and judicious; especially, in dealing with others
3. Subtle and circumspect, ensuring that no undue attention is attracted.
The word Discreet can be thought of as disecret. It can be thought of as keeping something secret and hence the meaning.
1. Her friends could always rely her to be discreet.
2. I made discreet inquiries about his credit rating.