Touch ID is Apple's biometric fingerprint authentication technology. With it, the Home button can unlock your iPhone or iPad, authorize your purchases on the iTunes Store or App Store, and use Apple Pay. A capacitive ring activates the scanner on contact which then takes a high-resolution picture of your fingerprint. That fingerprint is then converted into a mathematical formula, encrypted, and carried over a hardware channel to a secure enclave on Apple's A-series chipset. If the fingerprint is recognized, a "yes" token is released. If it's not, a "no" token is released.
In the perpetual battle between security and convenience, where many people would rather go without a passcode or strong password than fuss with anything complicated on mobile, Apple's Touch ID fingerprint identity sensor aims to do for authentication what iCloud did for backup and restore - make it easy enough that people will actually use it.