The film itself follows the story of a group of friends who are at a house party in New York. One of the characters is documenting the party using a camcorder, and it's this view of events that we watch as the audience. The idea is that this tape has been recovered by the US Government and kept as one of the few pieces of evidence of what happened to New York during the attack.
After around half an hour of fairly tedious party antics, in which we are introduced to all the characters and their backgrounds, things start to go pretty pear-shaped. Buildings are collapsing, the ground is shaking, there are some strange noises around and people are scared, panicked and in some cases just plain dead.
We then see our partygoers attempting to escape New York, all the while finding out details of events as the characters discover them for themselves. We see occasional, rushed glimpses of the monster, but mostly it's some serious shaky-cam that may well give you motion sickness.
I don't want to give away anything more about the film (although to be honest there isn't much to give away), but I will say that this one has left me with one overriding impression, and that is that Cloverfield is almost exactly like The Blair Witch Project. It has a cast of characters that we don't care about, we see all the events through their handheld camera, it's pretty much an exercise in suspense and the ending of the film will leave you with the same unsettled feeling.
However, I didn't rate Blair Witch as an especially good film, and I don't think Cloverfield is a good deal better. Watch it if you want to find out what all the fuss is about, but don't expect to be impressed when you do find out.