Going into any detail about
Crisis Core: Final Fantasy VII's story would only ruin the experience; what's important to know is that it follows the exploits of Zack Fair, the dark-haired hero whose life plays a not-so-small role in one of
Final Fantasy VII's more powerful character revelations. One of the more interesting aspects of the game's story is that it is primarily character-driven; there's no great, overarching plot involving the destruction of the world, but rather a small scale story focusing on the relationships between three SOLDIER operatives, Genesis, Angeal and Sephiroth, and how Zack gets sucked into the middle of a rivalry. The story has its confusing moments, though it really comes through in the end with a few brilliant storytelling touches that really make you sympathize with a certain character to the point that you might put the PSP down for a minute to think.
Since Crisis Core is a prequel, knowledge of VII's plotline isn't vital, but it wouldn't hurt to refresh your memory. At the very least, it makes some of the fan service moments more enjoyable. As expected, Sephiroth and Aerith play strong central roles, though you'll also run into a slightly happier (and more vocal) Cloud and Tifa.
Crisis Core's layout is similar to the opening section of VII. The Shinra Building serves as the main hub, though you have limited ability to explore the surrounding areas. Several missions will take you outside Midgar, though these areas are small and leave little room for exploration. Each section is featured in its own self-contained chapter, though these are more for story purposes, giving you the freedom to explore various side quests.
Aside from story-driven quests, you'll also uncover Missions, which are small side-quests that are accessed at Save Points. These are a big part of what makes Crisis Core a great PSP game. Story-based quests are lengthy, which isn't a good fit for a portable game, so Missions allow you to squeeze in some play time when you don't have that much. Missions are completely optional, though they are the only way to earn some items or some summons. The downside to missions is that there isn't much variation in mission types or locales. If you go through missions in large chunks, they tend to run together.