[Sprint Samsung Galaxy S 4 Q&A, Help & Troubleshoot] nobody291:
To me, one of the hardest parts of this whole thing is getting to the point where the basic terms make sense. I've been at this for a couple of years with an HTC Evo and now the GS4 since October, and sometimes it still makes my head spin. Here is an attempt to explain my understanding of a few things:

1. Booting...you can boot into Download mode, recovery, or system. Download mode is what you use with ODIN, you need the correct drivers on your computer (which can be found lots of places), so that ODIN and your phone can communicate. In download mode, the phone just accepts whatever ODIN sends it, so it's the easiest way to totally brick your phone. Don't mess with partitions and follow the instructions for whatever you're doing very carefully, because I think this is the only way to "hard brick" your phone.

2. Recovery - something you use to create images of your phone's ROM (nandroids), flash ROMs and mods, and restore past nandroids. The stock recovery will only flash approved updates from the carrier, so to do any of the things I listed you need a custom recovery. Most people use Philz (which is based on the ClockworkMod recovery) or TWRP. To flash the custom recovery, use ODIN and follow the instructions in the thread exactly. There is a way to flash recoveries without ODIN, but I've never done it. I was using TWRP, and switched to Philz to flash the rooted stock 4.3 update because the thread suggested it. They both seem to do the same thing, and I have been happy with both. I'm sure there are technical differences, but I couldn't explain what they are. You can switch back and forth as much as you'd like between the recoveries with ODIN. Recoveries can flash a complete ROM or just partial changes (like the hotspot mod). In a custom recovery (or the stock) you can also do factory resets and cache cleaning (most ROMs suggest you do this before flashing). Factory reset doesn't return the ROM to the factory original, it just clears out all of the user data (including user apps). A nandroid is the fastest way to restore a phone if you screw it up...I always make one before making significant changes to the phone. It takes several minutes to do, but I think it's well worth it. The operating system can't be running when you make one, that's why you need to boot into recovery mode. You create a Nandroid in Philz by selecting "backup and restore" then "backup to" which will allow backup to the internal storage or external SD card.

3. ROM - the actual operating system used by the phone when you actually use it as a smartphone. There are 2 basic flavors for the GS4...touchwiz and AOSP (Android Open Source Project). Touchwiz is Samsung's "flavor" of Android, AOSP is the "pure Google" version of Android. A lot of the things the GS4 will do rely on touchwiz (multiwindow is one example...but there are several). Some people really like the AOSP ROMs (Cyanogen is AOSP), but make sure you understand their limitations. I've never used one on the GS4. There are several touchwiz custom ROMs which have various features. I've tried a few and just settled on the stock rooted touchwiz ROM, but there are lots of options from great developers. I think most custom ROMs require you to use a custom recovery to flash them, but there might be ways to do it using ODIN. I like having a custom recovery, so I've only used ODIN for flashing modems and recoveries.

4. Firmware - as far as I can tell, there are two important parts of the firmware...the modem (sometimes called "radios" or "baseband"...find your version by looking in "about device"->"Baseband version"...the different modems are described by the last 3 letters you see) and the bootloader. The modem and the version of the ROM you flash need to match or you'll have problems with the radios in the phone (wifi seems to be the biggest problem if you have a mismatch). MF9 was the last modem version based on 4.2.2, and 4.3 has had MJA and MK2 (MK2 is the latest). The bootloader is what the phone uses to initially decide how to boot up. If you have the 4.3 version of the bootloader it will include "Knox", which will prevent you from flashing older versions of the modem, so you'll have to use a 4.3 ROM. Knox also "trips" a counter if you flash a custom ROM and Samsung claims they won't honor the warranty on a phone with the "Knox flag" tripped. If you have the 4.3 bootloader you're stuck with it (for now at least). I still have the 4.2.2 bootloader, so I don't have much more to offer on this subject. There are ways to get the 4.3 modem and ROM without the 4.3 bootloader. There is a lot here (and lots on youtube) on how to navigate the Knox minefield if you care about it. If you still have the 4.2.2 bootloader you can flash older (and newer) modems as much as you'd like without getting the updated bootloader using ODIN.

Hopefully this helps...this is a great place with lots of great people. Sometimes the scale of the amount of information here is overwhelming. But, search is your friend along with lots of time! Feel free to keep asking questions.