The Bible, in Genesis 1.27, tells us that God made man in His own image. That is a huge honor! But come on, what does being 'made in His image' even mean? Some people think it means we physically look like God; some (like Augustine) think it's that we're smarter than animals; and some think it's because we have a soul. But the real truth is in Ephesians 4.24: the Image of God (or Imago Dei) has to do with inward things like character, true righteousness, and true holiness. Since God is invisible Spirit (John 4.24) we get to be that visible, everybody-can-see-it demonstration of what God is like. That's the Imago Dei - we represent God's nature. We are kind of like God's ambassadors, if that helps you see it a little bit more clearly.
What It Looks Like In Real Life
If you want to know what it looks like to live out this doctrine, Ephesians 4.24-5.2 shows us: because we are made in God's image, we, unlike nature, get to represent God's truthfulness (put away all falsehood), his righteous anger (be angry and do not sin), his love for grace and building up rather than crass talk and tearing others down (let no corrupting talk come out of your mouths) and his absolute kindness and forgiveness towards us (be kind to one another... forgive one another, as God in Christ forgave you). After all that, St. Paul of Tarsus finished by writing 'Therefore be imitators of God'. We imitate God and live up to the Imago Dei by living out these kinds of principles/commandments in our lives. In fact, all of God's commandments help us live up to the Imago Dei; from commands against lying to instructions against gay marriage, all of the commandments have something to do with the Imago Dei (but we'll get to that later or else this post would become very very long).
Images, But Not Replacements
We represent God in the world, but we don't replace Him. He still judges the earth and lets nations rise or fall. He still gets involved on the earth by answering our prayers or choosing to grant faith to sinners. He does most of His work through us (proselytizing/evangelizing) and shows Himself to the world through us (ethics, marriage, culture, etc.), but God did not just create us and send us out so that He wouldn't have to stick around. As the Bible shows over and over again Yahweh (God) is still involved in our ongoing historical story and nothing happens apart from Him. We represent Yahweh (God) but we aren't standing in for Him as replacements. God is still involved, even if He chooses to do most of His work with or through us.
Summary & Further Reading
We were made to reflect God's nature to the world;
Our lifestyle is the representation of His character.
[EDIT:] This is an update of my recent post, Made In God's Image. My wife read the last post and told me that my longer posts are much better, and that I shouldn't stick to a firm word count. So I added a lot of material and this is the result. There are 29 more posts on the Imago Dei coming: some topics include Gay Marriage, Body Modification, The Trinity (three posts), Evangelism, and Social Justice. I'll try to get 1 or 2 up every week until they're all done.