Which is the closest exoplanet to Earth?
Earlier this year (2016), scientists used the radial velocity method to discover a planet orbiting Proxima Centauri: Proxima Centauri b. It was announced in Anglada-Escudé et al. (2016). Here are some of its basic properties, as reported by the authors of the paper and known as of August 2016:


Mass: At least 1.3 Earth masses
Semi-major axis: 0.05 AU
Maximum eccentricity: 0.35
Orbital period: 11.2 days
Surface temperature: 234 K (see Table 1)


Now, as the authors note, the planet may not be habitable:


  The habitability of planets like Proxima [Centauri] b—in the sense of sustaining an atmosphere and liquid water on its surface—is a matter of intense debate. The most common arguments against habitability are tidal locking, strong stellar magnetic fields, strong flares and high ultraviolet and X-ray fluxes; but none of these have been proved definitive.


However, further observations might help us learn more about the planet and determine if it could be habitable for any type of life.

It should be noted that the controversy about planets around Alpha Centauri continued after this question was initially answered by user8. Alpha Centauri Bb was shown to be nonexistent by Rajpaul et al. (2015). However, in another twist, observations by Demory et al. (2015) seemed to show the existence of Alpha Centauri Bc (at the time, Alpha Centauri Bb was still a possibility, so now "Alpha Centauri Bc" would actually be "Alpha Centauri Bb" if it existed), an Earth-like planet. However, there has been no more evidence to confirm the existence of this planet - and at any rate, Proxima Centauri is closer to the Solar System than Alpha Centauri, so Proxima Centauri b wins no matter what.