About Bitcoin Scripts
Scripting languages date back from thousands of years ago. Mathematic formulae are an example of scripting language.
Bitcoin scripts use a specialized scripting language to write secure smart contracts and complex transactions.
A Stack-based Scripting System
Bitcoin uses a scripting system for complex transactions. Forth-like, Script is stack-based: instructions are 1-byte opcodes processed from left to right to be pushed on top of the stack sequentially, left-most first. All values are variable length byte arrays. Script is not Turing-complete, with no loops, for optimal security.

A portfolio of scripts
There are many use cases for Bitcoin scripts.
Stay tuned: you'll find here links to some more demos.
Lightning Network: a compelling use case for Bitcoin scripts
The Lightning Network white paper ( draft version 0.5.9.1 dated November 20, 2015) describes a decentralized network for scalable, efficient payment channels. Transactions are sent over this routing network of micropayment channels whose transfer of value occurs off-blockchain. Bitcoin scripts enable time locked and hash locked transactions so that the payment protocol between users, via any number of Lightning Network hubs, remains trustless. This white paper by Joseph Poon and Thaddeus Dryja is probably the most important contribution since the invention of Bitcoin.
Download White Paper