I have confidence in confidence alone.

"You're really good-looking," my wife said to our third son.
"I know," he answered.
Overconfidence is sometimes confused with self-esteem and vice versa. Our son isn't being arrogant. He feels confident about his looks and that's that. Kids can't have excessive self-esteem, but they can certainly be arrogant. Julie Ross, M.A., describes self-esteem as internal and that it refers to a sense of confidence and competence, whereas arrogance points to overconfidence that's out of touch with reality and which has external manifestations, such as bragging about something. Actually, she explains, bragging can point to a lack of self-esteem.
I titled this post with a quote from the lyrics to The Sound of Music to emphasize that self-esteem is key when setting out to do something bold. That's what is underlying in the scene in which Maria is walking towards the Captain's house for the first time. The character of Maria points to her physical appearance in the last line as misleading with regards to her true self-esteem.
We can get clues about how confident children are about themselves by looking at their actions. Confident children are willing to take on a challenge and accept the risk of failure. Our role as parents is to provide sufficient opportunities to attain goals. This demands avoiding overprotection and reassuring our children. Confidence, self-esteem, or self-assurance are undeniably necessary. They are so, because our children will have to go out and do things without our supervision. They'll get hurt, they'll fail more than once, but it'll be their "confidence in me" that will get them on their feet.
If you're having doubts as to where to start offering opportunities, start at home. Ask your children to take on a task traditionally done by you, like make a bed, heat something in the microwave, water the garden/flowers, use the dishwasher. Gradually ask your children to go to the corner store on his own or pay the fair on the bus or subway. Give them a list at the supermarket for them to fetch. When the time comes for your children to take on something more complex, their confidence will be your confidence too.