Right now it's hard to imagine all the use cases for smart contracts because the technology is so new and developers are still exploring what can be done with them. It's a little like when the Internet started out, people weren't sure what it was good for and couldn't imagine all the amazing stuff that would come later like Facebook, Instagram, Twitter, etc.
Ethereum's smart contracts are essentially bits of computer code that execute on the blockchain. They are implemented in a programming language called Solidity, which is Turing complete. By Turing complete, I mean that Solidity is a full featured language which can implement any computer program that your imagination can dream up (although whether that program would be efficient and practical to run on a blockchain is another matter altogether).
And that's the advantage Ethereum has over Bitcoin. It can do everything Bitcoin can as far as being a store of value is concerned, but it also has a programming language built into it. Imagine Bitcoin as a simple calculator, then Ethereum is Excel with macros support.
Some example uses of smart contracts include: automated escrow for asset exchanges, locking up money to be released at some specific date in the future (like a trust fund), proof of ownership of physical assets (Digix gold), decentralized exchanges (Etherex), decentralized prediction markets (Augur), gambling apps (Rouleth), and many more that have yet to be dreamed up. Right now the first generation of dapps are kind of hard to use and require technical knowledge of Ethereum, but in the future they will be as simple to use as going to a web site and ordinary people will have no idea that Ethereum is the underlying technology used to implement them, just as when you go to Facebook you don't need to care about the technical details of whatever database system is being used under the covers.