Method 1 (Tethered Jailbreak):

Step 1: Download iOS 6.1.3 from here: IPOD TOUCH 4G
Step 2: Download iOS 6.0 from here: iOS 6 (iPod Touch 4G): Link
Step 3: Download RedSn0w 0.9.15b3 from here: Windows / Mac.

Step 4: Connect your device to your computer, and place it into DFU mode.

Step 5: Open RedSn0w 0.9.15b3, and ensure it detects your device in DFU mode at the bottom of the windows.

Step 6: Click Extras > Select IPSW > and select the iOS 6.0 IPSW that we told you about in step 2. RedSn0w should successfully identify the iOS 6.0 firmware. If not, you have the wrong version of RedSn0w, or the wrong firmware. Click OK.

Step 7: Click back, and then click Jailbreak. Once everything finishes loading, ensure that Install Cydia is checked, and click Next. The jailbreak will start, assuming you placed your device into DFU mode correctly as asked in Step 4.

Step 8: Once your device is back to the Lock screen, place it back into DFU mode.

Step 9: Click OK on the Tethered Jailbreak notification if you have not already done so, and again click Select IPSW.

Step 10: Again, select the iOS 6.0 firmware for your device. Once it successfully identifies the firmware, click Just boot.

Step 11: If you successfully placed your device into DFU mode, and followed all previous steps correctly, RedSn0w will tether boot your device. You should see the infamous pwnapple logo as your device boots, which indicates you were successful.

Step 12: Once back at the Lock screen, unlock the device, and swipe to the second page. Once there, you should see Cydia. 

Method 1 (Untethered): Not Confirmed.