It's not really an MMO.
The basic idea is that you create a hunter (create the look, pick weapon specialties, etc.) and take on contracts to hunt down giant monsters. Different weapon specialties lead to different play styles, and there's also tons of traps and bait you can use against your prey.
The real draw (for the PSP series) is the co-op multiplayer. I've never gotten a chance to do it myself (I have Freedom 2 for PSP), but co-op multiplayer in Monster Hunter is apparently very, very good, because it really forces you to work together to succeed.
Monster Hunter is massively popular over in Japan, and is one of the big reasons that PSP is still a popular system over there. It's one of the system's "killer apps." The formula for the game is so successful that Capcom is basically replicating it in Lost Planet 2.
I'm definitely planning on getting Tri, even though I'm kind of bad at Freedom 2 (though that's largely due to my inability to deal with the PSP game's kind-of-clunky controls). The game looks awesome and it comes from a strong franchise.