Supported Devices for iOS 6
iPhone 3GS, iPhone 4, iPhone 4S, iPhone 5, iPod Touch (4th and 5th generation), iPad 2, iPad (3rd and 4th generation) and iPad Mini
Apple rolled out the iOS update from 6.1.2 to 6.1.3. Initially the thought was that within a matter of weeks the evad3rs jailbreak would be updated as well, but as of June 12th there is no untethered jailbreak available for 6.1.3. There are many sites claiming to have the update but those sites are purely promoting the 6.1.0 update as a 6.1.3 jailbreak.
There is a tethered jailbreak for iOS 6.1.3 which can be done via Redsn0w, see here
To jailbreak the iPhone 5 it has now become more difficult as the iPhone 5 has now been updated to version 6.1.4, but again, the most recent jailbreak is still only for version 6.1.2.
To jailbreak a version 6.1.2 you can visit Evad3rs to Jailbreak your iOS 6.1.2.
This video shows how to jailbreak iOS6 version 6.1.2
Another option is to use the Redsn0w method to jailbreak iOS5 or iOS6. See here