Just buy it, already.
Without spoiling anything, I'll just say that the story is great. Seeing the events that led up to FF7 is one of the biggest highlights of this game, while those who aren't familiar with the original will still be able to get very involved in the story. Still, the game's plot will probably have a bit more gravity to those who've already played FF7, or at least those who have an understanding of that game's plot.
Crisis Core's graphics are another standout feature. This could easily be mistaken for a PS2 game by anyone who didn't already know what system it was on, and a pretty good looking PS2 game at that. The CG cutscenes look as good as anything in Advent Children, and the battles that take place are just as frantic. Sound also plays a major part in Crisis Core. Most of the music is remixed from FF7, so it all sounds fantastic. The voiceovers and dialouge are equally great.
Gameplay outside of battle is classic Final Fantasy. You only have one character in your party this time, but you can still equip him with new items and skills (although you can never change which weapon is equipped, and many of the items you get will be worse than the ones already equipped.) You can also combine items and materia to make new ones. Battles play out in real time. You can switch between skills such as magic, physical attacks, and item usage with the shoulder buttons, and execute the maneuver with X. It feels very similar to Kingdom Hearts. Your most powerful skills are regulated by a spinning slot wheel on the top of the screen. When three images line up, you'll perform a powerful attack or even a summon. When three numbers line up in a certain way, you'll be granted status effects such as invincibility. When three sevens line up, you'll level up.
Combat is a bit of a departure from the Final Fantasy games of old, but it's a lot of fun nonetheless. It's very simple, so that anyone can pick it up, and very fast paced and slightly random so that it stays fun for the 40+ hours you'll need to beat the game. Still, there are some enemies who are really stupid and just stand there eating your sword until the battle ends. If you want a real challenge, play on Hard mode, but I prefer the more easygoing Normal difficulty myself. Bosses still provide challenges on Normal.
Overall, the way that the excellent story, graphics, sound, and gameplay meld together make Crisis Core feel like a true numbered FF game on the PSP. It is quite simply one of the best gaming experiences out there on any platform.