The official iOS 6.1 Evasi0n jailbreak tool has been released for iOS devices. Evasi0n will jailbreak all iOS devices capable of running iOS 6.0-6.1, including the iPhone 5 and iPad mini. The only current device that is not supported is the third-gen Apple TV.
The Evad3rs, a group of prominent iOS hackers behind past jailbreaks, have released their free tool for iOS 6.1 online. There are Mac, Windows, and Linux versions.