How do you jailbreak your device on iOS 6?
Compatibility

Thankfully, this jailbreak is supported by all version of iOS 6, including iOS 6.0, 6.0.1, 6.0.2, and 6.1. It works on all devices that support iOS 6 and it won't magically make iOS 6 work on devices that can not upgrade to iOS 6. Some current devices that run iOS and are not compatible with this one jailbreak are the 3rd generation Apple TV and the 2nd generation Apple TV.

Evasi0n modifies the OS for the following hardware:


iPhone 3GS
iPhone 4
iPhone 4S
iPhone 5
iPod Touch 4th Generation
iPod Touch 5th Generation
iPad 2
iPad (3rd Generation) with Retina display
iPad (4th Generation) with Retina display and Lightning connector
iPad Mini


You need a computer to run the jailbreak. As for those operating system requirements, Evasi0n is compatible with Windows (XP minimum), Mac OS X (10.5 minimum) and Linux (x86 / x86_64).

Set Up

First, I suggest updating your device to iOS 6.1, as it is the latest version of iOS, and the most secure. Now, apparently you can not update via Wifi. You must do it in iTunes. If you already updated wirelessly,  you can [SHIFT + Update] on Windows or [Option + Update] on Mac OS X, in iTunes.

Some notes:

According to the official Evasi0n ReadMe file you should also ensure that you backup your device before Jailbreaking, disable the passcode lock if enabled and ensure iTunes does not launch during the Jailbreaking process.

Backup your device using iTunes (or iCloud) before using evasi0n. If something breaks, you’ll always be able to recover your data.
Those who use backup passwords in iTunes must disable them for now. After doing so, iTunes makes a brand new backup. Please wait for that backup to complete before proceeding! Feel free to re-enable your backup password after jailbreaking.
Please disable the lock passcode of your iOS device before using evasi0n. It can cause issues.
Launch evasi0n, plug in your device, and click “Jailbreak”. Just sit back and observe its progress. Watch for any steps you may be asked to perform.
Avoid all iOS and iTunes related tasks until evasi0n is complete.
If the process gets stuck somewhere, it’s safe to restart the program, reboot the device (if necessary by holding down Power and Home until it shuts down), and rerun the process.


Detailed Walkthrough
(From iJailbreak.com)


The first thing you will need to do is download Eavsi0n for Windows, Mac OS X or Linux. Just head over to evasi0n.com for that.
With Evasi0n downloaded you will need to extract it. On Mac OS X you will find a .dmg file that upon being clicked you will be able to drag the files to your desktop, but one Windows you will have a .zip file that can be extracted by right clicking on it and then clicking the extract to folder option.
After the Evasi0n Jailbreaking tool has been extracted find the Evasi0n app / .exe file. On Windows you will need to right click on it and then click the Run as administrator option. On Mac OS X 10.8 Mountain Lion and above you will need to hold down the control button on your keyboard, click the Evasi0n app and then the Open option to get past Gatekeeper.
With Evasi0n up and running you will need to plug your iPhone, iPod Touch or iPad into your computer via the USB cable (upon doing this Evasi0n should recognize your device). Before you proceed any further you will also need to ensure you disable your lockscreen passcode if you have one enabled. This can be done by launching the Settings app, going into the General tab, followed by Passcode Lock and then tapping Turn Passcode off.
With your lockscreen Passcode disabled, iPhone, iPod Touch or iPad plugged into your computer and iTunes closed you can click the Jailbreak button. The process will now begin and you should see the progress bar moving forward.


Sit back and do not touch your computer until Evasi0n tells you to unlock your device and search for a newly created icon by the name of Jailbreak.
When you see this message simply follow its instructions.


Unlock your iPhone, iPod Touch or iPad
Find the newly created Jailbreak icon on your homescreen
Tap this icon ONCE, it will launch and quickly close itself. This is normal.


You will then see the Evasi0n Jailbreaking tool start back up again and the progress bar should start moving. When the Jailbreak is complete you will see a “Done!” message displayed . This means the Jailbreak was successful! You will now just need to sit back until your iOS device automatically restarts, which might take a several tries.
When your device restarts it will display a white background with an Evasi0n logo. This is again part of the Jailbreaking process and you will see messages like Setting up Cydia, Clearing SpringBoard cache etc.



Once everything is finished being configured your device will be brought to the lockscreen. In which case you can unlock your device, find the Cydia icon on your SpringBoard and be on your way!
You should now have a Jailbroken iPhone, iPod Touch or iPad. Congratulations! If you are experiencing troubles then my only suggestion would be to restart your computer, switch USB ports and try again. If you still don’t have any luck then a factory restore on iOS 6.1 may be necessary before reattempting. Also keep in mind that you cannot have a Passcode lock enabled.
Once you are Jailbroken you might be asking yourself, what now? Well, you will want to launch Cydia and install tweaks of course. Make sure you check out our Top Cydia tweaks and top Siri tweaks section to find some of the best tweaks available.

Congrats:

You're done! Don't forget to install OpenSSH, as you might need it (In case of troubles). Instructions should be found on the Cydia Store. Have fun!

NOTE: This originally came from my blogpost, http://infinite-josiah.blogspot.com/2013/02/how-to-jailbreak-your-ios-6-device.html