The Earth is round because it is in hydrostatic equilibrium: its gravity is strong enough, and the material comprising most of its bulk fluid enough, to make it flow into the shape that minimizes its gravitational potential energy: a sphere.
As a fairly reasonable analogy, you can think of the Earth as behaving like a floating drop of water; while the forces that hold the Earth and the drop together are somewhat different (gravity for the Earth, surface tension for the water drop), the overall behavior is similar. The important thing to realize is that, when it comes to the overall shape of the planet, what matters is the gravity and the hydrostatic properties of the core and the mantle; the relatively thin solid crust on the surface is neither thick enough nor solid enough to make any significant difference.
Once you appreciate that, it should be obvious what would happen if a large piece of the Earth were to magically disappear somehow: the remaining bulk of the planet would just flow into the hole, filling it. Once things settle down, you'd just end up with a slightly smaller, but still spherical, Earth.
Now, that's the long-term outcome. The slightly more interesting question is, what would happen in the short term? For that, I'll have to agree with the other answers: utter chaos and destruction. Basically, as bowlturner notes, cutting a large chunk out of the Earth would be very similar to, and only slightly less destructive, than hitting it with a large planetoid — like, say, the one that produced the Moon.
Basically, here's a number of things that would happen:
The atmosphere will flow down into the hole. This will actually take several hours, since the wind speeds will only(!) rise to around Mach 1. Indeed, the initial rarefaction wave will only travel at Mach 1, so on the opposite side of the Earth, it will take a while for the air to start moving at all. Still, in less than a day (if nothing else interferes; see below), pretty much all of the atmosphere should be down in the hole, leaving the rest of the planet (temporarily) airless.
The oceans will also flow down into the hole. The speed at which this happens will depend somewhat on the location of the hole (but probably not as much as you'd think; see below). Sound travels about five times faster in water than in air, so the initial disturbance will circle the globe pretty fast, but the actual quasi-equilibrium flow rate may (initially) depend on undersea geography. In any case, you'll likely end up, briefly, with some of the biggest waterfalls ever seen on Earth (and there have been some pretty big ones).
As the water flows down into the hole, it will hit exposed magma and vaporize, joining the air that is also flowing into the hole. This will also cool the surface of the magma very efficiently, forming a thin solid crust, but since the magma will also be moving (see below), that crust is unlikely to be very stable.
Depending on geography, and the extent of crustal damage (see below), some standing bodies of water might be left in lakes and oceanic trenches. If so, as the atmosphere above them disappears down the hole, the pressure will drop until liquid water becomes unstable. At that point, the water will boil and freeze at the same time; the part that boils will spread out as vapor (and, if it doesn't condense as frost, eventually end up in the hole with the rest of the atmosphere), while the rest will remain behind as solid ice. (If the ice gets thick enough, some liquid water might remain, trapped underneath it.)
The lithosphere will also flow down into the hole. Remember, I said that the Earth's mantle will flow under gravity until it becomes roughly spherical again. As the mantle flows, it will drag the crust along with it — but since the crust is solid (on short timescales, anyway), it's going to break up as it flows, much like ice on moving water. The size of the chunks is hard to estimate, but I'd expect them to range from dozens of kilometers (comparable to the thickness of the crust) up to maybe a few thousand (comparable to small tectonic plates today) in diameter.
Besides causing the mother of all earthquakes, the breakup of the Earth's crust will open deep fissures all over the planet. That's why I said that undersea geography may not matter so much — many of these fissures will open up under oceans (where the crust is thinner anyway), some likely along existing mid-ocean ridges, and the water will just flow into them and hit the exposed magma there.
The details of what happens next will depend on many things, like the depth of the water and the spreading rate of the fissure. Initially, at least in the deep ocean, the water can probably absorb the heat from the emerging magma, as it does in ordinary undersea volcanic eruptions, but as the fissures keep spreading and the ocean keeps draining, eventually the remaining water may get hot enough to boil away.
So, over the first few minutes to hours after the hole forms, what we'll see (hopefully from a safe and comfortable vantage point, say, on Mars), is basically a massive avalanche of magma, water and air all rushing in to fill the hole. Without running an actual lithodynamic simulation, it's hard to say exactly how long it will take for this avalanche from hell to reach the middle of the hole (or, rather, to fill it up — of course, the magma will also be flowing up from the bottom of the hole), but when it does, it's likely to do something interesting.
It will splash.
Basically, all that magma, water and air will accumulate a considerable amount of momentum as it rushes into the hole. Once it reaches the center, and meets the flow coming in from the other side, it has nowhere to go but up. And up it goes. Basically the same effect happens when you drop a rock in water: at first, it just basically makes a hole in the water, just like your magic hole in the Earth, but most of the splashing actually happens only when the water falls back into the hole.
Again, it's a bit hard to estimate the magnitude of the splash off the top of my head (though an experienced planetologist perhaps might), but I'd expect it to scatter at least some magma into the sky. Some might even escape the Earth's gravity entirely, but most will likely just fall back somewhere else on the planet. So, just after you thought it couldn't possibly get any worse, any surviving bits of the Earth's surface will likely experience a rain of burning rocks from orbit.
Finally, as the dust settles (literally), the remaining atmosphere of the planet (probably somewhat thinned by the splash, but augmented by a whole lot of water vapor and volcanic gases) will gradually settle down again. Most of the water vapor (which used to be oceans) will rain down again, at least once the surface cools down below the boiling point of water, and eventually the clouds will break up and the sun will shine on the surface once more.
(At least, that's if you're lucky, and all the extra heat, water vapor and volcanic outgassing isn't enough to kick off a runaway greenhouse effect like on Venus. We don't really know enough about climate dynamics at such extreme conditions to say if such an equilibrium shift would even be possible on Earth, or if so, what would be needed to trigger it.)
OK, at that point, you may wonder what all the is going to do to any life there might be on the planet. Well, obviously, the simple answer is "nothing good". Indeed, I would expect all this destruction to kill off well over 99.99% of all life forms on the planet — what doesn't get crushed or burned or boiled might end up vacuum-frozen or just plain poisoned by toxic gases.
Still, life on Earth is pretty tenacious, and absolutely everywhere. There are microbes that live inside solid rock, deep within the Earth's crust, and others that can tolerate anything from freezing to boiling. Short of actually melting the entire crust, I doubt it would be possible to exterminate all life on Earth (although a runaway greenhouse state as mentioned above, if possible, might just about do it).
Depending on the extent of the disruption, it's even possible that some multicellular organisms might survive — perhaps even some macroscopic land animals like insects, some of which might survive as eggs or pupae. I'm not seeing much hope for vertebrates, though, but if you really wanted to stretch plausibility, I just might buy a few fish surviving in some lucky refuge somewhere, perhaps in a deep frozen lake. That would be a pretty long shot, though.