Wow, I want my PSP to do that! Have any good links?
I'll see if I can find them.
First you'll need a modded PSP, with m33 Firmware (might work with others, but that's what I've been using). Older CFW actually uses an emulator, and handles PSX games more crappily.
If you want to be able to use your TV as a screen for standard PSP games and you want it to not suck, you'll need FuSa (and a psp2000 or 3000)
If you want to be able to play PS1 games on your TV using your PSP, you don't even need FuSa, just the adapter cable.
Here's a link to a tutorial that explains the process, and has a bunch of handy links to the apps you'll need! 
(They may be outdated though)
And the newest version of
Popstation.
Newest version of
PSX2PSPAnd here's some
technical stuff about how some converters don't handle some games very well, and saying which ones to use if you have problems.
There are some sites online where you can get the games, already converted; but I never bothered with that much, since pretty much all the games I'd want I have a PSX Disc For.
ff6 on PSP and ff7 on PSP were fun ways to kill time on the bus. Oh; on the PSP you can pause the game any time by pressing the home button iirc.
Umm. Multi-Disc Games are a bit harder; you're better off hunting for pre-combined isos that put all the discs into one psp eboot.
You can compress the PSX Games, and I did notice FFT was slightly slower when I had the game on the highest compression (It took very little space on my memory card though.)
And here are some pre-made
graphics/sounds for your PSP Menu for the games.
And
HereAnd some
other ones for non PSX Games.
This one's really Nice for
FF6I can't seem to find a good one for FFT, though you could extract the menu from WotL I suppose.
Do you think they could've ported FFT to another language, and that might cause the slowdown?
I don't think so, I mean why bother. Basically all they have to do is change the screen resolution to match the psp screen, and maybe one or two changes for the psp speakers, and then keep in mind they have a better processor, better graphics chip, more RAM, and that UMDs can hold more than CDs to work with, so they're free to add more stuff.
But yeah; you can pretty easily run any of your PSX Games, or Modded PSX Isos on your PSP.
If you test FFT on your PSP this way, let me know if there is a comparable slowdown.
Also, if you have M33 firmware, at the main menu, press select, and change your USB device to UMD Disc, you can now rip the iso off the UMD, put it on your PC, then when you change it back again, you can transfer it back to your memory card. It runs faster from your memory card, in my experience, and personally I didn't notice the slowdown on mine until someone pointed it out; but as I said, I never played it from the UMD.
Apparently many of the PSX Games can be bought on the PSN if you want to buy them over again for your PSP. Through from what I understand, most of them haven't been changed at all to fit the PSP better, and are exactly the same as a PS1 Rip.
