Asking of the British Museum, "What's it like?", is roughly analagous to saying, "Hey, the history of the world, what's it like?" This museum is daunting and you must, MUST, take at least two days to appreciate it. If you try to run through it in a few hours, you're going to get absolutely nothing out of it and just end up exhausted and overwhelmed.
Don't try to narrow it down to highlights, either. You may walk through the main entrance saying "First the Rosetta Stone, then the Elgin Marbles, then...," but I guarantee that you'll see the Easter Island statues in the lobby and stop for those. Then on your way toward the Elgin Marbles, you'll stop and meander through another exhibit, and another. By all means, walk in with a general plan of things you MUST see, but the best idea is just to wander and get overwhelmed by the things you had no idea were there.
That said! The bowling-over starts on the outside. The building itself is impressive enough, especially with the striking half-face statues in front. Now go ahead, walk through the doors,and, WHAM, you're in the Great Court, the largest, covered public square in Europe. Look up at the ceiling - it's striking.
From here, go where you will, but some of my personal favorites:
The Elgin Marbles - these make the highlights lists, sure, but to see them in person is really something. These are bits of the Parthenon, from the frieze. Take some time to examine the detailing - it's breathtaking. And once you're done seeing the Marbles, learn a bit about the controversy surrounding them and their place in the museum instead of Greece, a modern-day colonialist conflict.
The Egyptian Collection - this entire section is engrossing. It doesn't matter if you don't know anything about Egyptian history, because this exhibition will teach you what you need to know. This is the history of one of the most powerful civilizations ever to grace the face of the earth, laid out in a museum wing. Oddly, the Rosetta Stone, for all of its historical significance, doesn't have a whole lot of impact in person - I walked past it three times before I realized what it was, and I was LOOKING for it.
The British Isles, Viking, and Celtic Exhbits - I ended up spending more time in this area the second day than the first. This is Stone- and Bronze-Age history laid right out for you, in artifact after artifact. Especially interesting, I thought, was a large, tiled panel featuring an intricate design (see photo).
All in all, the British Museum makes a "Brief History of Time" under one roof, an experience to remember. Wander through slowly, take your time, and you won't regret it.