What is a sidechain?
Put simply, sidechaining is any mechanism that allows tokens from one blockchain to be securely used within a completely separate blockchain but still moved back to the original chain if necessary. By convention the original chain is normally referred to as the "main chain", while any additional blockchains which allow users to transact within them in the tokens of the main chain are referred to as "sidechains". For example, a private Ethereum-based network that had a linkage allowing ether to be securely moved from the public Ethereum main chain onto it and back would be considered to be a sidechain of the public network.
@mowliv I think a good way to think about it is by looking at our economy. The Federal Reserve prints US dollars for the US Government (the main blockchain) to boost the US economy. However, US dollars can be exported to other countries (a side chain) that could have a completely independent economy but still use a currency backed by the US government.
@Olshansk not a great example for lots of reasons. A better example would be: if a doge-ethereum linkage was built that allowed doge users to exchange their doge for a specific erc20 token on ethereum and vice versa. There is actually a bounty out there for this one.