Ok let's start off simple.
In Persona 3 FES for the Playstation 2, you can use the emulator PCSX2 and a feature known as hostFS, or host filesystem, to make it so that you don't have to repack the ISO (the file that the game comes in) every time you want to install a mod.
To set up hostFS we are going to need a few things. First on that list is a version of PCSX2.
While the newer version of PCSX2 boasts cool features like texture replacement and stuff like that, they also practically removed the feature to use hostFS without some trickery, so if you want to play Persona 3 FES and don't really care about texture mods (or simply want to develop your own mods), I would recommend downloading the "Stable" version of PCSX2 (1.6.0) for the foreseeable future.
Now that we have our emulator, it's time to get a bios. There are two ways to do this, the first and easiest way is to go on archive.org and search for "PS2 Bios" and download one of those. The second method is to extract a PS2 bios from the PS3 bios you can download online, I won't go into it but there's a great video on YouTube all about it.
Once you've run PCSX2 and put the bios in the bios folder and selected it in PCSX2, close out PCSX2 completely and go into the "inis" folder of your PCSX2 folder and open the file "PCSX2_vm.ini" in a texteditor like Notepad and find the line where it says
Next we need our ISO file of Persona 3 FES. The best way to get this is either archive.org again or another online source. Once you have the ISO open that bad boy in 7zip and extract the file named "SLUS_216.21" to the same folder as your ISO and rename it to "SLUS_216.21.ELF"
Now open PCSX2 and open the menu option that's called CDVD, go to where it says "ISO Selector" and click Browse and select your ISO file. Now go to the System option and click Run Elf... and select the elf file you set up in the same folder as your iso.
For more information and info on how to set up everything legally and with mods and more, check out p3f.cep.one
EDIT: ALSO YOU NEED THIS PNACH FILE!
Place it in your cheats folder and make sure to enable cheats in PCSX2