Nandroid Backup on Android is refer to a name for a full backup of an Android device. A full Nandroid backup will backed up or contains everything of the Android content even to the smallest and invisible things to the user from data, system, kernel, modem, etc. So, say something bad is happen to your Android device like bootloop and softbrick after apply mods, used a root risky application, or installed a custom ROM, we just need to restore from the last Nandroid backup we made. As far as I can recall, Nandroid backup has saved my Android phone life to escape from such bad situations like bootloop and soft brick many times more than my fingers number :D
To create a Nandroid backup for your Android device is very easy because today it is available as one of "standard" feature I think, on every custom recovery. So just install any popular custom recovery available for your Android device, and its almost sure it will have Nandroid backup option (in the recovery mode, that feature may will be named only as "Backup"). And, my preference of best custom recovery for Android I think is Team Win Recovery Project or TWRP (just in case you ask my opinion, hehe).
Know it's very important and very usefull, so it is recommended to always make a Nandroid backup before doing some mods or hack job to our Android device system. Nandroid backup is also recommended before we installing or using a powerfull root application like GlTools, Lucky Patcher, Xposed, etc. So if the application installation is make your Android device bootloop or even soft brick, no need to worry because we still have the second life of our Android device: The Nandroid backup.
Make a Nandroid backup using CWM or TWRP recovery on your Android deviceWell there are some of custom recoveries available for Android out there, but there is just two of most popular and most used, it is ClockworkMod (CWM) and Team Win Recovery Project (TWRP).
To make a Nandroid backup of your Android device using CWM recovery, the option is available on "Backup and Restore" menu. On TWRP, the option is available on "BACKUP" menu. Different on CWM, on TWRP recovery we can even choose invidually what partitions we want to backup and the preferred storage (internal or external storage).
Note that if you are using an Android device which has feature, and, already moving applications to the external SDCard, the application will be moved to directory /storage/externalSdCard/.android_secure/ (hidden directory).
In CWM recovery, the /storage/externalSdCard/.android_secure/ directory and the internal storage will also be backed up while in TWRP is not. So the Nandroid backup created using CWM is likely will have bigger size compared the one which created using TWRP recovery.
And before make a Nandroid backup, don't forget about one thing to prepared, it is space. Like we talked above, Nandroid backup is about a backup which can contain every data even to the smallest thing from data partition to system partition. So I am sure a Nandroid backup size we will create later will have a huge sizes, it's likely will have more than 500 MB or even Gigas.
To calculate how much a Nandroid backup size will take, just see your Android device internal storage. How much your device storage has been eaten, and the Nandroid backup size is likely will not less from that. Just in case, you should prepare a little more space than the expected size.