NASA's twin STEREO probes are each a quarter of the way around the sun, heading in different directions, and as of yesterday, they're positioned to collectively be able to see the entire surface of the sun, all at once, incidentally proving that the sun is in fact round and not just a flat disc, which is what it looks like. Oh well.
The probes are equipped with extreme ultraviolet imagers (extreme!) to take pics of the sun itself, coronagraphs to image the sun's corona, and heliospheric imagers to keep track of the space between the sun and the Earth which sometimes gets filled with exciting coronal mass ejections. Their opposing orbits will take them around the back of the sun, where they will cross paths and start back towards Earth, during which time the Solar Dynamics Observatory in Earth orbit will pick up the slack. All in all, we'll have this 360 degree view for at least eight years.
There are a couple reasons that it's important for us to be able to see the backside of the sun. Occasionally, the sun gets pissed off and ejects a couple billion tons of plasma into space at a million miles an hour. When these coronal mass ejections hit Earth, they can fry satellites and astronauts and even cause power outages on the ground. We obviously can't stop these things from hitting us, but being able to see them coming is important too, and until now, if the sun was all sneaky and fired something off around the back, we'd be clueless until it was too late to do much. Also, as we start colonizing other planets, we'll need space-weather warning systems for then, too.
The other good reason to be able to see the whole sun at once is to figure out what makes it tick. Since the sun is made up of plasma, which is sort of like a liquid, when something big and nasty happens in one place, it can send shockwaves all the way around the surface and make other things happen in other places. When we could only see one part of the sun at once, it was hard to tell just how all of these events were connected, but now we should start getting a much better idea of how and why the sun does what it does.
Below, we've got some video from NASA further explaining how it all works, plus a bunch of pictures from the STEREO spacecraft.