Let's say you have multiple mods, such as texture replacers, that replace the same texture (Iron Sword for the sake of the argument). You start up your game but find that the texture for Iron Sword is not the one you want. Typically, you would load the plugin (.esp file) for the texture mod you want - let's call it Mod A - after the plugin for the texture mod you don't - Mod B - and the problem should resolve itself.
What if Mod A or Mod B doesn't have a plugin that you could reorder? What if neither do? How do you tell Mod Organizer (MO) to prioritise Mod A over Mod B?
The first thing you need to do is ensure which mod is taking priority over Mod A's textures.
1. Open Mod Organizer, and on the left hand pane, look for Mod A. You should see a lightning symbol with a plus and/or minus symbol, under a column titled "Flags". The plus symbol means that Mod A is taking priority over other mods, the minus symbol means that other mods are taking priority over Mod A. For our example, Mod A should have a minus symbol, and Mod B should have a plus symbol.
[Note: If you hover the mouse pointer over the lighting symbol, the UI will show a tooltip stating one of three things: "Overwrites files" for the plus symbol, "Overwritten files" for the minus symbol and, "Overwrites and Overwritten" for both. Mod Organizer doesn't actually overwrite any files, hence Mod Organizer needs to set priorities for the mods you use, and decides which files from which mods to use.]
2. Double left click on the symbol for Mod A, and you will see a window pup up.
3. Click on the conflicts tab. You will see two windows. The window on top ("The following conflicted files are provided by this mod") lists the files that MO uses from this mod over others. The window at the bottom ("The following conflicted files are provided by other mods") are the files that MO uses from other mods that replace the files from this one. In our example, you would see the texture file for Iron Sword from Mod B in the bottom window.
4. Close the window.
5. On the left hand pane, there should be a column titled "Priority". What you want is for Mod A to have a higher priority than Mod B. Note that the higher the number, the higher the priority.
6. Single left click on the number in the priority column for Mod A and you will be able to change the number. Set the number of Mod A higher than that of Mod B. So, for example, if Mod B had a number 10, then you need to set Mod A to 11 or higher. Don't worry if there's already a mod with a priority of 11, MO will automatically update the priority numbers accordingly.
7. Click the refresh button on the left pane (small button to the right of the Profile drop down menu), to refresh the list. The changes should then be apparent.
8. Repeat step 2 and 3, and check that the texture file for Mod B does not appear in the window below.
9. Do the same for Mod B, and check that the texture file from Mod A appears in the window below.
That's it, you should be ready to go. Start up your game and check that the correct texture file got loaded.
Hope this helps!
[color=#ff8c00;]This information is technically all in the wiki on the priorities tab, but still is a valuable read for advanced info relating to hidden files and such. It does go off-topic later on. I may expand wiki priorities information to be more informative for specific examples like this in future.[/color]