I just want to start this answer off by saying that if you are not an expert computer user, this is probably not for you. You will need to know how to setup and edit partition tables, along with knowing how to resize, delete and add partitions. I of course accept no responsibility for any damage you cause to any device from reading this tutorial.
In theory here is what you need to do:
Repartition your device's SD card so that you can make room for all the the Operating Systems you want to run
Get a new boot loader that supports dual (or multi) booting
Flash the partition with a build that is compatible with your phone
Specifically for your device (the Galaxy Note 10.1), here is a GREAT link on dual booting:
The tutorial is on an Ubuntu machine, if you do not have Ubuntu it is pretty easy to install from their website:
You probably don't even need to install Ubuntu to do this tutorial, I think you would be fine just running off of the live CD. I am a little biased because I love Ubuntu, but I would recommend installing it when you have free time. It really is a life saver and it runs on top of a linux kernel so you get all of the benefits of linux when you use Ubuntu.
VERY IMPORTANT BEFORE YOU START THE TUTORIAL
1) Backup your device every time before you make a change to the OS or to the boot loader. It is extremely easy to brick your device if you don't know what you're doing. If you brick your device and you have no backups to go from, then you are out of luck my friend.
2) You need to have a CWM (Clockwork Mod) installed on your device before you start the tutorial along with Nandroid Backup. See Extra Stuff below for install information.
3) I cannot stress this more: MAKE SURE THAT YOU ARE INSTALLING A MOD THAT FITS PERFECTLY WITH YOU DEVICE!! You will potentially render your device unusable if you do not check your ROMs thoroughly.
4) This is not so important but will save you a lot of time in the future. Go get a bigger SD card, no 1GB or 2GB is not enough and 4GB is pushing it. Go out and buy a 16GB SD card, they are not that expensive and if you are going to be doing multiple booting, then you will want the extra space. It will be a huge pain to transfer all of your files onto a bigger SD card because you decided you wanted to triple boot Android, Arch Linux and Ubuntu all on one device. This is the voice of experience unfortunately.
Hopefully the tutorial works out well for you. When you get comfortable with installing OS partitions, then you can start trying out different flavors of Linux. My favorite is Ubuntu, followed closely behind by Arch Linux (ARM) (Which is great for server applications).
Here is a Video tutorial for installing the CWM on the Galaxy Note 10.1 if you need to do that. Again make sure you get the right ROM for your device.
If YouTube is more your thing, then here is a pretty good video on doing the same thing in the other tutorial. This YouTube video requires you to already have CWM installed on your device.
For those of you who do not have the Galaxy Note, here is something else I found that might be useful (or at least a starting point):
Please feel free to edit if you have something to add (Tutorials for specific devices etc.)
Also, if you need more info, just comment and I will help you out.