Encouraging independance in our children is very important. It can be too easy to do everything for them. But for them to learn sometimes we need to take a step back and say "You try". I like to ask my children if they need help. So they know I am willing to help if they need it, yet I have confidence in their ability to try.
I love the age between 18mths and 3 years. This is when I have seen my big kids learn and grow so much. They  are wanting to do things for them selves. "Me do"
My big boy is almost 4 and even when he needs to go to the toilet he will say, "Mum I need to go to the toilet" So I will say, "You need some help, come on then" I am almost offended when he will say, "No I don't need help!". But I am very proud. And when he gets dressed he will tell me off if I help with the buttons. "Mum, I can do it!" And even if he doen't get the buttons in the buttonholes, it makes me feel proud. I have to smile at him in his wonky pjs. Ahhhh he even tells me off if I help him clean up his toys. "They are my Thomas toys! "
Encouraging independance also encourages self esteem. When they do something all by themselves, even as babies  learning to walk or crawl, they feel a sense of acheivement. And all we have to do as parents is smile!
We of course can more actively encourage our children by making things 'easy' to start off with. Such as buying shoes with velcro so they can put on their own shoes. And providing challenges when things are too easy for them. Such as asking them to turn off the tv when they haven't done yet. The first time I asked my boy, he said "Which one?" And boy was he proud when he 'magically' turned it off.
Encouraging our children to do things even the littlest things helps them realise their potential and their worth. It also helps them see where they fit into the family. As they grow they can also have responsibilty. This again helps their self esteem.
We were visiting a friend on the weekend and my girl was talking non stop. Our friend said, "Its great - she is brimming with confidence" I think that is my main objective - to help my children be asure of themselves and not need me for everything.