The Alpha Centauri system. Source
The Alpha Centauri system is easily seen with the naked eye, however, you'll only see one bright point of light if you go looking for it, even though it is a binary system (two stars orbiting each other). The reason for this makes sense, the stars orbit each other at a distance that changes between the distances of Pluto to the Sun and Saturn to the Sun. That's fairly close, but still far enough apart that one could distinguish the two stars with something as common as a pair of binoculars.
Alpha Centauri is a binary system. Source
The closest star to our solar system is Proxima Centauri at a mere 4.24 light years away, however, this star is a part of the Alpha Centauri system. Alpha Centauri is 4.37 light years away in the same direction as Proxima, and it seems like Proxima is gravitationally tied to the binary system. All this means that Alpha Centauri is the closest star system to us. Even more awesome, the two binary stars are very similar to our sun. Alpha Centauri A is about 10% more massive and 23% larger than our sun. It is even of a similar color. Alpha Centauri B, the companion star to A, is a little smaller. It is only about 90% as massive as our sun, and 14% smaller. However, both stars have a very similar composition to our sun, sparking interest in the possibility of finding planets like Earth orbiting one or both of these stars.
A screen shot from Sid Meier's Alpha Centauri based Civilization game. Source
Alpha Centauri is widely seen as the first destination for interstellar space flights. The chemical composition of the system increases the likelihood of finding earth-like planets, and its proximity to our own solar system makes it the easiest system to get to. Because of these factors, considerable resources have been put into finding planets orbiting these stars. So far we have been able to determine that there are no gas giants in the system, but budget setbacks have hindered the search for smaller planets. One advantage of finding a lack of gas giants is that it increases the chances of finding terrestrial planets that might harbor life.
One of the biggest fears surrounding the search for these planets is that having two stars so close to each other might make the accretion of any planet impossible. And even if a planet can form, there's no guarantee that water could ever reach it. In our solar system the gas giants diverted water bearing comets from the outer Oort cloud into the inner solar system where they landed on earth, supplying us with that all-too-necessary fluid. In the Alpha Centauri system, however, there doesn't seem to be a surrounding Oort cloud to supply water bearing comets. The good news is that computer simulations show that a planet could have a stable orbit around Alpha Centauri A for 250 million years, and it is possible for either Alpha Centauri A, B or even C (Proxima) to divert comets into the inner solar system.
In a few years, we should be able to detect planets as small as 1.8 earth masses orbiting around either star in Alpha Centauri, so let's hope that we find something interesting.