We report on the \textit{Fermi}-LAT observations of the Geminga pulsar, the second brightest non-variable GeV source in the $\gamma$-ray sky and the first example of a radio-quiet $\gamma$-ray pulsar. The observations cover one year, from the launch of the $Fermi$ satellite through 2009 June 15. A data sample of over 60,000 photons enabled us to build a timing solution based solely on $\gamma$ rays. Timing analysis shows two prominent peaks, separated by $\Delta \phi$ = 0.497 $\pm$ 0.004 in phase, which narrow with increasing energy. Pulsed $\gamma$ rays are observed beyond 18 GeV, precluding emission below 2.7 stellar radii because of magnetic absorption. The phase-averaged spectrum was fitted with a power law with exponential cut-off of spectral index $\Gamma$ = (1.30 $\pm$ 0.01 $\pm$ 0.04), cut-off energy $E_{0}$ = (2.46 $\pm$ 0.04 $\pm$ 0.17) GeV and an integral photon flux above 0.1 GeV of (4.14 $\pm$ 0.02 $\pm$ 0.32) $\times$ 10$^{-6}$ cm$^{-2}$ s$^{-1}$. The first uncertainties are statistical and the second are systematic. The phase-resolved spectroscopy shows a clear evolution of the spectral parameters, with the spectral index reaching a minimum value just before the leading peak and the cut-off energy having maxima around the peaks. Phase-resolved spectroscopy reveals that pulsar emission is present at all rotational phases. The spectral shape, broad pulse profile, and maximum photon energy favor the outer magnetospheric emission scenarios.Read less <