Often times when buying a console from a third party, it will come with a parental control set. On an RGH/JTAG, you have the option to look up a code to reset it or modify your NAND to remove the code. On an unmodified console, you have a few official options outside just trying to guess the code.
A video demonstration of bypassing the family settings can be found on MrMario2011's channel.
Unmodified Console
On unmodified consoles, your options to remove a parental lock are as follows:
Official Reset Code Retrieval
This method has reportedly been discontinued, but is kept here for posterity.
Open the Devices page for your Microsoft account, register your console, select the console, then select More Actions > Reset Passcode. If this isn't possible, you can call Microsoft's customer support, and they will provide you with a reset code, along with the instructions on how to use it.
If the correct code is entered, you will be prompted to reset your Xbox 360 settings. Select Yes. This will reset all settings but will not delete any of your saved games or other data. If you did not receive this prompt, enter the entire button combination again from the beginning.
Guessing the Recovery Question
Navigate to Family Settings > Family > Content Controls, enter the wrong code, and it will prompt you to reset the passcode. It will ask you a password reset question. Try and guess the answer if you can.
Guessing the Factory Reset Code
Navigate to System Settings > Console Settings > System Info. Press (don't hold) the following button combination: 
followed by four more buttons or D-Pad directions out of the list below:
- + ANY
DIRECTION 
-
-
- ANY DIRECTION +
- ALL FOUR DIRECTIONS
If the correct code is entered, you will be prompted to reset your Xbox 360 settings. Select Yes. This will reset all settings but will not delete any of your saved games or other data. If you did not receive this prompt, enter the entire button combination again from the beginning with a different code ending.
RGH/JTAG/Bad Update
Using Parental Control Remover
Using Parental Control Remover — which can be downloaded directly to your console with the Homebrew Store or manually downloaded from here and copied to your console—you can launch the app and view the currently set passcode, view the system settings reset code, or press to turn off parental controls and clear the passcode.
Looking up your reset code in a NAND Dump
- Obtain a dump of your NAND if you don't already have one.
- Open J-Runner and select the "Load Source" button and choose your NAND dump. Select Tools > SMC Config Editor. Note the four letters in the field to the left of "Reset Code".
- Navigate to System Settings > Console Settings > System Info. Press (don't hold) the following button combination: followed by the buttons corresponding to the four letters. If the correct code is entered, you will be prompted to reset your Xbox 360 settings. Select Yes. This will reset all settings but will not delete any of your saved games or other data. If you did not receive this prompt, enter the entire button combination again from the beginning.
Removing the Parental Lock by Flashing the NAND
- Obtain a dump of your NAND if you don't already have one.
- Open J-Runner and select the "Load Source" button and choose your NAND dump. Select "Advanced XeBuild Options" and check the
nomobilebox. Press OK to close the window. - Select "Create Image". Rename the file to
updflash.binand place it on a USB storage device and plug it into your console. Boot into XeLL; it should automatically find the update and flash it. - Reboot the Xbox 360, and the new NAND image should be applied with parental controls disabled.
Arduino Brute Force
The last option is to automatically brute force the password reset code using an Arduino and [[File:XBOX360_ParentalCode_Reset.zip|ParentalCode Reset by Agent24] (source site). This may take up to 15 hours to run, depending on how quick it can find the correct code.