Context. The radio galaxy IC 310 has recently been identified as a γ-ray emitter based on observations at GeV energies with Fermi-LAT and at very high energies (VHE, E > 100 GeV) with the MAGIC telescopes. Originally classified as a head-tail radio galaxy, the nature of this object is subject of controversy since its nucleus shows blazar-like behavior. Aims: To understand the nature of IC 310 and the origin of the VHE emission, we studied the spectral and flux variability of IC 310 from the X-ray band to the VHE γ-ray regime. Methods: The light curve of IC 310 above 300 GeV has been measured with the MAGIC telescopes from 2009 October to 2010 February. Contemporaneous Fermi-LAT data (2008-2011) in the 10-500 GeV energy range were also analyzed. In the X-ray regime, archival observations from 2003 to 2007 with XMM-Newton, Chandra, and Swift-XRT in the 0.5-10 keV band were studied. Results: The VHE light curve reveals several high-amplitude and short-duration flares. Day-to-day flux variability is clearly present (>5σ). The photon index between 120 GeV and 8 TeV remains at the value Γ ~ 2.0 during both low and high flux states. The VHE spectral shape does not show significant variability, whereas the flux at 1 TeV changes by a factor of ~7. Fermi-LAT detected only eight γ-ray events in the energy range 10 GeV-500 GeV in three years of observation. The measured photon index of Γ = 1.3 ± 0.5 in the Fermi-LAT range is very hard. The X-ray measurements show strong variability in both flux and photon index. The latter varied from 1.76 ± 0.07 to 2.55 ± 0.07. Conclusions: The rapid variability measured in γ-rays and X-rays confirms the blazar-like behavior of IC 310. The multi-TeV γ-ray emission seems to originate from scales of less than 80 Schwarzschild radii (for a black hole mass of 2 × 108 M⊙) within the compact core of its FR I radio jet with orientation angle 10°-38°. The spectral energy distribution resembles that of an extreme blazar, albeit the luminosity is more than two orders of magnitude lower.