Sol Trigger is in Japanese aside from some of the menus and character names having been fan-translated, so I can't say that what I gathered of the story from playing it is 100% accurate (in fact I can promise you it isn't), but after reaching the end I feel as if I got the gist of things.
The game takes place in the city of Kaiserhald, where the eponymous resistance group 'Sol Trigger' fights a secret war against the Machine Church who rule everything. The Machine Church uses a power source called Sol to power everything from streetlights to horrible robot monstrocities, but what the general populace doesn't know is that Sol is extracted from living people, and thus the reason for the resistance.
You play as Farel, who leads Sol Trigger after the initial leader is killed during a mission, and embark on a series of operations to disrupt the Machine Church's harvesting and manufacturing of Sol. Things obviously get more complicated than that, but I'd like to avoid anything to spoiler-y.
The cutscenes in the game are amazingly well done
Sol Trigger handles very similarly to games from the Shin Megami Tensei series, with a first person view during battle and a system that maps out the fairly convoluted dungeons as you explore them. Where it differs, however, is what makes the game so great. Animation quality for just about every ability is superb, and I found myself using abilities that may not have been optimal just because they looked cooler. Abilities even change appearance once you've leveled them up to their cap, which gives a bit more incentive to varying how you play.
Another unique aspect of the game is that you can use your parties Sol (basically MP/SP) to power up their abilities. Say ability A cost 50 Sol to use normally, you could power it up to use 75, 100, 200, etc. and the damage or effects it did would vary accordingly. Very cool, and it allows for a much greater range of control over how you handle your battles.
One negative I will say is that there are two characters in both halves of the game that are so game breakingly useful that you're almost forced to use them in your party 100% of the time despite always having a decent sized group to switch in and out. Cyril with her group wide defense/attack buff and heal over time along with Sophie/Wilma's ability that eliminates the Sol cost of the groups other attacks are just leagues better than most of the alternatives, so not using them feels like deliberately handicapping yourself.
Despite the first person battle system, abilities all have really nice looking animations
If you can get past the fact that the game is in Japanese, I'd say this is easily one of the best JRPGs on the PSP. It managed to keep me interested throughout and there were even some genuinely emotional moments here and there. Definitely pick this one up if you can.