To really understand this, you would need to read the protocol specs. But I'll describe it briefly.

Each transaction contains a script. In order to claim the bitcoins from a transaction (e.g.: you want to spend them in a new transaction) you have to provide whatever proof is described by the script.

Normally, the script is pretty much the same: the intended recipient has to provide a digital signature that proves they own the destination address. But you can requires a lot of different kinds of proof: require more signatories, no proof at all, or require the claimant to solve a puzzle.