KIC 8462852, or the WTF (Where's the Flux?) star, is an F-type main sequence star about 1,480 ly away.  It's a little larger and more massive than the sun, and a few times brighter.  Age is uncertain, but probably older rather than younger.  

Kepler observations over the last few years reveal very strange large and aperiodic flux variations (up to 20%) - of the general form predicted by some ETI megastructure models.  However there doesn't appear to be any excess infrared.

The star's fluctuations were discovered by the PlanetHunters team.  In the WTF paper they review a large number of unlikely natural explanations and settle on an unusual comet swarm as the most likely scenario.

Abstract of the WTF paper:

Over the duration of the Kepler mission, KIC 8462852 was observed to undergo irregularly shaped, aperiodic dips in flux down to below the 20% level. The dipping activity can last for between 5 and 80 days. We characterize the object with high-resolution spectroscopy, spectral energy distribution fitting, and Fourier analyses of the Kepler light curve. We determine that KIC 8462852 is a main-sequence F3 V/IV star, with a rotation period ~0.88 d, that exhibits no significant IR excess. In this paper, we describe various scenarios to explain the mysterious events in the Kepler light curve, most of which have problems explaining the data in hand. By considering the observational constraints on dust clumps orbiting a normal main-sequence star, we conclude that the scenario most consistent with the data is the passage of a family of exocomet fragments, all of which are associated with a single previous breakup event. We discuss the necessity of future observations to help interpret the system.

 

From "Comets or Aliens?", on the Planet Hunters blog: " However, so  far over 100 professional scientists have had a look at the lightcurves and not managed to come up with a working solution."

In a another recent paper Jason Wright et al discusses the WTF star in more detail and critiques the comet theory.

The Search for Extraterrestial Civilizations with Large Energy Supplies. IV: the Signatures and Information Content of Transiting Megastructures:

Arnold (2005), Forgan (2013), and Korpela et al. (2015) noted that planet-sized artificial structures could be discovered with Kepler as they transit their host star. We present a general discussion of transiting megastructures, and enumerate ten potential ways their anomalous silhouettes, orbits, and transmission properties would distinguish them from exoplanets. We also enumerate the natural sources of such signatures. 
Several anomalous objects, such as KIC 12557548 and CoRoT-29, have variability in depth consistent with Arnold's prediction and/or an asymmetric shape consistent with Forgan's model. Since well motivated physical models have so far provided natural explanations for these signals, the ETI hypothesis is not warranted for these objects, but they still serve as useful examples of how nonstandard transit signatures might be identified and interpreted in a SETI context. Boyajian et al. 2015 recently announced KIC 8462852, an object with a bizarre light curve consistent with a "swarm" of megastructures. We suggest this is an outstanding SETI target. 
We develop the normalized information content statistic M to quantify the information content in a signal embedded in a discrete series of bounded measurements, such as variable transit depths, and show that it can be used to distinguish among constant sources, interstellar beacons, and naturally stochastic or artificial, information-rich signals. We apply this formalism to KIC 12557548 and a specific form of beacon suggested by Arnold to illustrate its utility.

Jason Wright discusses WTF here on his blog.

Big reddit discussion on r/askscience here.